TITLE
Dissertation submitted to
Centre for Research and Post Graduate Studies in Mathematics
Ayya Nadar Janaki Ammal College, Sivakasi
Include ANJAC logo
in partial fulfilment of the requirements for the award of the degree of
Master of Science in Mathematics
by
Naveen Kumar U
(Registration No. 21PM34 )
Under the Guidance of
Dr. P. Gnanachandra
Assistant Professor
Centre for Research and Post Graduate Studies in Mathematics
Ayya Nadar Janaki Ammal College
(Autonomous, Affiliated to Madurai Kamaraj University, Re-accredited with ‘A+’ grade in the 4th cycle by NAAC with
CGPA 3.48 out of 4, College of Excellence by UGC, Star College by DBT and Ranked 83rd position at National Level in
NIRF 2022)
Sivakasi - 626 124
MAY-2023
Dr. C. Parameswaran, M.Sc., M.Phil., B.Ed., Ph.D.,
Associate Professor and Head,
Center for Research and Post Graduate Studies in Mathematics,
Ayya Nadar Janaki Ammal College (Autonomous),
Sivakasi - 626 124.
Certificate
This is to certify that this project report entitled ”AN INVESTIGATION
OF THE PROBLEMS FROM THE GROUND OF INTEGER GEOMETRY”
being submitted by Mr. K. Parthasarathy (Reg. No: 21PM36), final year student of M.Sc. degree course in Mathematics, Ayya Nadar Janaki Ammal College (Autonomous), Sivakasi, affiliated to Madurai Kamaraj University, Madurai, is a bonafide
record of work carried out by him under the guidance and supervision of Dr. KM.
Kathiresan, Assistant Professor, Department of Mathematics, Ayya Nadar Janaki Ammal College (Autonomous), Sivakasi.
It is further certified that this project report has been scanned for Plagiarism
using URKUND software available in the college library and the level of Plagiarism is
found to be less than the level recommended by the Academic Integrity Committee of
the college.
Sivakasi
28 April 2023
Signature of the
HOD
(C. Parameswaran)
1
E-mail: parames65 c@yahoo.com
Dr. KM. Kathiresan, M.Sc., M.Phil., Ph.D.,
Assistant Professor,
Department of Mathematics,
Ayya Nadar Janaki Ammal College (Autonomous),
Sivakasi - 626 124.
Certificate
This is to certify that this project report entitled ”” being submitted by
Mr. K. Parthasarathy (Reg. No: 21PM36), final year student of M.Sc degree programme in Mathematics, Ayya Nadar Janaki Ammal College (Autonomous), Sivakasi,
affiliated to Madurai Kamaraj University, Madurai, is a bonafide record of work carried
out by him under my guidance and supervision.
It is further certified that to the best of my knowledge, this project report or
any part thereof has not been submitted in this college or elsewhere for the award of any
other degree or diploma.
Sivakasi
April 2023
Signature of the Guide
(KM. Kathire-
san )
E-mail: jayram.kannan@gmail.com
2
Mr. K. Parthasarathy,
II M.Sc. Mathematics (Reg. No: 21PM39),
Center for Research and Post Graduate Studies in Mathematics,
Ayya Nadar Janaki Ammal College (Autonomous),
Sivakasi - 626 124.
Declaration
This is to certify that this project report entitled ”AN INVESTIGATION
OF THE PROBLEMS FROM THE GROUND OF INTEGER GEOMETRY”
has been carried out by me in the Center for Research and Post Graduate Studies in
Mathematics, Ayya Nadar Janaki Ammal College (Autonomous), Sivakasi, affiliated to
Madurai Kamaraj University, Madurai, in partial fulfillment of the requirements for the
award of the degree of Master of Science in Mathematics.
I further declare that this project report or any part thereof has not been submitted
in this college or elsewhere for the award of any other degree or diploma.
Sivakasi
April 2023
Signature of the Student
(K. Parthasarathy)
E-mail: parthasarathyk@anjaconline.org
Reg. no.: 21PM39
3
Acknowledgement
“Nantri marapathu nanranru”
-Thiruvalluvar.
First of all I thank the Almighty God, for giving me the knowledge and strength to complete this work successfully.
I am deeply grateful to The Management, Ayya Nadar Janaki Ammal College, Sivakasi,
for providing me all the facilities required for the completion of this work.
I wish to thank Dr. C. Ashok, Principal, Ayya Nadar Janaki Ammal College, Sivakasi,
for his valuable support and guidance.
I extend mt gratitude to Prof. R. Jaganathan, Associate Professor and Head, Department of Mathematics(U.G.) for his guidance and motivation.
My sincere thanks to Dr. C. Parameswaran, Associate Professor and Head, Centre
for Research and Post Graduate Studies in Mathematics, Ayya Nadar Janaki Ammal
College, Sivakasi, for his guidance in the pursuit of the study.
I would like to express my deep and sincere gratitude to my project guide
Dr. K.
M. Kathiresan for his continuous support.
I wish to express my sincere thanks to all the staff members of the Department of Mathematics, Ayya Nadar Janaki Ammal College, Sivakasi, for their constant help from the
beginning to till end.
Finally, I heartly express my gratitude to my family members and friends for their encouragement to me on various occasions.
(K. Parthasarathy)
4
Contents
1
Introduction
1
2
Preliminaries
2
3
2.1
Matlab Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
2.2
Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Application of matlab in testing certain operators
5
3.1
Self-adjoint operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
3.2
Normal operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
3.3
Quasinormal operators . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
3.4
Quasi P normal operators . . . . . . . . . . . . . . . . . . . . . . . . . .
19
3.5
n power normal operators . . . . . . . . . . . . . . . . . . . . . . . . . .
22
3.6
K-Quasinormal operators . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
3.7
K-Quasi P normal operators . . . . . . . . . . . . . . . . . . . . . . . . .
29
5
4
3.8
n power Quasinormal operators . . . . . . . . . . . . . . . . . . . . . . .
32
3.9
n power K-Quasinormal operators . . . . . . . . . . . . . . . . . . . . . .
37
3.10 n power K-Quasi P normal operators . . . . . . . . . . . . . . . . . . . .
42
Programs II
47
4.1
r+ic type operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
4.2
2r*3ic operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
5 Conclusion
66
Reference
67
6
1
Introduction
This project comprises of various MATLAB programs to determine a variety of properties
of various bounded linear operators of a Hilbert space H(say).
Programming language MATLAB was developed primarily for matrix calculations, hence
the name ”MATLAB” (Matrix Laboratory), and it plays a significant part in a variety
of scientific disciplines. MATLAB® introduced a feature called scripts, including live
scripts, starting in R2016b. These scripts can contain code to define functions. These
processes are commonly referred as local functions. If you wish to reuse code within a
script, local functions can be helpful. We can easily determine whether a property is true
for various operators by developing functions for various operator properties.
Using MATLAB’s script function, we have created a number of programmes for this
project that examine various features of bounded linear operators defined on a Hilbert
space H.
In chapter 2 (Preliminaries), we have introduced some properties of operators required
to proceed with the project’s main content.
In chapter 3 (Application of MATLAB in Testing Certain Operators), we have matlab programmes for each property stated in chapter 2 (Preliminaries), along with their
methods and example outputs.
In chapter 4 (), we have written a different set of MATLAB programs that enable us to
find different properties of some special operators. Such as the operators having entries
r + ic and 2rx3ic, where r and c are row and column indices of the operators respectively.
Then chapter 5 (conclusion), concludes the project and finally there is the reference
section for the project.
1
2
Preliminaries
2.1
Matlab Introduction
Matlab was initially known as the matrix programming language, the term ”MATLAB”
is an acronym for ”Matrix Laboratory”. It is a multi-paradigm, MATLAB. Therefore, it
can be used with a variety of programming paradigms, including functional, Visual, and
Object-Oriented calculations.
The built-in functions of MATLAB offer top-notch resources for performing calculations,
including optimization, linear algebra, numerical solution of ordinary differential equations (ODEs), data analysis, quadrate, signal processing, and many other scientific tasks.
The user can create their own functions in the MATLAB language. Thus they are not
restricted to using only the built-in functions.
The various uses of MATLAB are:
1. Developing algorithms
2. Performing linear algebra that is linear
3. Graph plotting for larger data sets
4. Data visualization and analysis
5. Numerical Matrix Computation
and many more.
2
2.2
Definitions
Definition 2.2.1. Self-adjoint operator Let H be a Hilbert space. Let L(H) be the
algebra of all bounded linear operators defined in H. Let T be an operator in L(H). The
operator T is called Self-adjoint if it satisfies the following condition T = T ∗
Definition 2.2.2. Normal operator Let H be a Hilbert space. Let L(H) be the algebra
of all bounded linear operators defined in H. Let T be an operator in L(H). The operator
T is called normal if it satisfies the following condition T ∗ T = T T ∗
Definition 2.2.3. Quasinormal operator Let H be a Hilbert space. Let L(H) be the
algebra of all bounded linear operators defined in H. Let T be an operator in L(H). The
operator T is called Quasinormal if it satisfies the following condition T (T ∗ T ) = (T ∗ T )T
Definition 2.2.4. Quasi P normal operator Let H be a Hilbert space. Let L(H) be the
algebra of all bounded linear operators defined in H. Let T be an operator in L(H). The
operator T is called Quasi P normal if (T + T ∗ )(T ∗ T ) = (T ∗ T )(T + T ∗ )
Definition 2.2.5. n power normal operator Let H be a Hilbert space. Let L(H) be the
algebra of all bounded linear operators defined in H. Let T be an operator in L(H). The
operator T is called n power normal if T n T ∗ = T ∗ T n for n > 1.
Definition 2.2.6. K-Quasinormal operator Let H be a Hilbert space. Let L(H) be the
algebra of all bounded linear operators defined in H. Let T be an operator in L(H). The
operator T is called K-Quasinormal if T (T ∗ T )k = (T ∗ T )k T for k is any fixed positive
integer.
Definition 2.2.7. K-Quasi P normal operator Let H be a Hilbert space. Let L(H) be
the algebra of all bounded linear operators defined in H. Let T be an operator in L(H).
The operator T is called K-Quasi P normal if (T ∗ T )K (T + T ∗ ) = (T + T ∗ )(T ∗ T )k for k
is any fixed positive integer.
3
Definition 2.2.8. n power Quasinormal operator Let H be a Hilbert space. Let L(H) be
the algebra of all bounded linear operators defined in H. Let T be an operator in L(H).
The operator T is called n power Quasinormal if T n (T ∗ T ) = (T ∗ T )T n for n > 1.
Definition 2.2.9. n power K-Quasinormal operator Let H be a Hilbert space. Let L(H)
be the algebra of all bounded linear operators defined in H. Let T be an operator in
L(H). The operator T is called n power K-Quasinormal if (T ∗ T )k (T n ) = (T n )(T ∗ T )k for
n > 1 and k is any fixed positive integer.
Definition 2.2.10. n power K-Quasi P normal operator Let H be a Hilbert space. Let
L(H) be the algebra of all bounded linear operators defined in H. Let T be an operator
in L(H). The operator T is called n power K-Quasi P normal if (T ∗ T )k (T n + (T ∗ )n ) =
(T n + (T ∗ )n )(T ∗ T )k for n > 1 and k is any fixed positive integer.
4
3
Application of matlab in testing certain
operators
3.1
Self-adjoint operators
Aim: To Create a program that shows if a given operator is self-adjoint or not.
Algorithm
1. The algorithm defines two variables, ‘w‘ and ‘wconj‘, and initializes them to some
value.
2. The algorithm prompts the user to enter an operator.
3. The algorithm assigns the input operator to the ‘w‘ variable.
4. The algorithm uses the ‘conj‘ function to obtain the complex conjugate of the ‘w‘
variable.
5. The algorithm compares the ‘w‘ variable with the ‘wconj‘ variable.
6. If the two variables are equal, the algorithm uses the ‘fprintf‘ function to print a
message that includes the value of ‘w‘ and the value of ‘wconj‘. The message could
be something like ”for, the operator is equal to w = ” followed by the value of ‘w‘
and the value of ‘wconj‘.
7. If the two variables are not equal, the algorithm uses the ‘fprintf‘ function to print a
message that includes the value of ‘w‘ and the value of ‘wconj‘. The message could
be something like ”for, the operator is not equal to w = ” followed by the value of
‘w‘ and the value of ‘wconj‘.
5
8. The algorithm ends.
Program
heading="To check if a given operator is self adjoint or not\n";
prompt="Enter the operator\n";
result1="The operator is self adjoint\n";
result2="The operator is not self adjoint\n";
fprintf(heading);
w=input(prompt);
wconj=conj(w)’;
if w==wconj
fprintf(result1);
disp("for,");
disp(w);
disp("is equal to ");
disp(wconj);
else
fprintf(result2);
disp("for,");
disp(w);
disp("is not equal to ");
disp(wconj);
end
6
Now let us run the program an operator to see whether the given operator is self-adjoint
or not.
Example 1:
To check if a given operator is self adjoint or not
Enter the operator
[4 3+8i 2-7i 18+9i 6+5i;3-8i 6 7-9i 18+2i 79-1i;2+7i 7+9i 19 -27i 0;
18-9i 18-2i 27i 27 -19i;6-5i 79+1i 0 19i 43]
4.0000 + 0.0000i
3.0000 + 8.0000i
2.0000 - 7.0000i
18.0000 + 9.0000i
6.0000 + 0.0000i
7.0000 - 9.0000i 18.0000 + 2.0000i
6.0000 + 5.0000i
3.0000 - 8.0000i
79.0000 - 1.0000i
2.0000 + 7.0000i
7.0000 + 9.0000i
19.0000 + 0.0000i
0.0000 -27.0000i
0.0000 + 0.0000i
18.0000 - 9.0000i
18.0000 - 2.0000i
0.0000 +27.0000i
27.0000 + 0.0000i
0.0000 -19.0000i
6.0000 - 5.0000i
79.0000 + 1.0000i
0.0000 + 0.0000i
0.0000 +19.0000i
43.0000 + 0.0000i
4.0000 + 0.0000i
3.0000 + 8.0000i
2.0000 - 7.0000i 18.0000 + 9.0000i
6.0000 + 0.0000i
7.0000 - 9.0000i
18.0000 + 2.0000i
7.0000 + 9.0000i
19.0000 + 0.0000i
0.0000 -27.0000i
6.0000 + 5.0000i
3.0000 - 8.0000i
79.0000 - 1.0000i
2.0000 + 7.0000i
0.0000 + 0.0000i
18.0000 - 9.0000i
18.0000 - 2.0000i
0.0000 +27.0000i
7
27.0000 + 0.0000i
0.0000 -19.0000i
6.0000 - 5.0000i
79.0000 + 1.0000i
0.0000 + 0.0000i
0.0000 +19.0000i
3.0000 + 8.0000i
2.0000 - 7.0000i
18.0000 + 9.0000i
6.0000 + 0.0000i
7.0000 - 9.0000i
18.0000 + 2.0000i
7.0000 + 9.0000i
19.0000 + 0.0000i
0.0000 -27.0000i
43.0000 + 0.0000i
The operator is self adjoint
for w,
4.0000 + 0.0000i
6.0000 + 5.0000i
3.0000 - 8.0000i
79.0000 - 1.0000i
2.0000 + 7.0000i
0.0000 + 0.0000i
18.0000 - 9.0000i
18.0000 - 2.0000i
0.0000 +27.0000i
27.0000 + 0.0000i
0.0000 -19.0000i
6.0000 - 5.0000i
79.0000 + 1.0000i
0.0000 + 0.0000i
0.0000 +19.0000i
3.0000 + 8.0000i
2.0000 - 7.0000i
18.0000 + 9.0000i
6.0000 + 0.0000i
7.0000 - 9.0000i
18.0000 + 2.0000i
7.0000 + 9.0000i
19.0000 + 0.0000i
0.0000 -27.0000i
43.0000 + 0.0000i
is equal to w*
4.0000 + 0.0000i
6.0000 + 5.0000i
3.0000 - 8.0000i
79.0000 - 1.0000i
2.0000 + 7.0000i
0.0000 + 0.0000i
18.0000 - 9.0000i
18.0000 - 2.0000i
0.0000 +27.0000i
0.0000 -19.0000i
8
27.0000 + 0.0000i
6.0000 - 5.0000i
79.0000 + 1.0000i
0.0000 + 0.0000i
43.0000 + 0.0000i
Example 2:
To check if a given operator is self adjoint or not
Enter the operator
[1 2 3;4 5 6;7 8 9]
1 2 3
4 5 6
7 8 9
1 4 7
2 5 8
3 6 9
The given operator is not self-adjoint
for w,
1 2 3
4 5 6
7 8 9
is not equal to w*,
1 4 7
2 5 8
3 6 9
9
0.0000 +19.0000i
3.2
Normal operators
Aim:To create a Matlab program to determine if the given operator is normal or not.
Algorithm:
1. Read the operator from the user and store it in the variable w.
2. Convert the operator w to its complex conjugate w’.
3. Multiply the operator w by w’ to get the operator wwconj.
4. Multiply the operator w’ by w to get the operator wconjw.
5. Check if the operator wconjw is equal to the operator wwconj.
6. If the operator wconjw is equal to the operator wwconj, then print the message
”The given operator is normal””for W X W* is equal to W* X W” and display the
operators wconjw and wwconj.
7. If the operator wconjw is not equal to the operator wwconj, then print the message
”The given operator is not normal” ”for W X W* is not equal to W* X W” and
display the operators wconjw and wconjw.
I. Program:
heading="To check if the given operator is normal or not\n";
prompt="Enter the operator\n";
result1="The operator is normal,\n";
result2="The operator is not normal\n";
10
fprintf(heading);
w=input(prompt);
wconj=w’;
wwconj=w*wconj;
wconjw=wconj*w;
if wwconj==wconjw
fprintf(result1);
disp("for W X W*");
disp(wwconj);
disp("is equal to W* X W");
disp(wconjw);
else
fprintf(result2);
disp("for W X W*");
disp(wwconj);
disp("is not equal to W* X W");
disp(wconjw);
end
Now let us run the program an operator to see whether the given operator is Normal or
not.
11
Example 1:
To check if the given operator is normal or not
Enter the operator
[4 3+8i 2-7i 18+9i 6+5i;3-8i 6 7-9i 18+2i 79-1i;2+7i 7+9i 19 -27i 0;18-9i 18-2i 27i 27
The operator is normal,
for W X W*
1.0e+03 *
0.6080 + 0.0000i
0.9180 + 0.5760i
-0.1040 + 0.3540i
0.3120 + 0.4890i
6.8090 + 0.0000i
0.0710 + 0.2240i
0.4960 + 1.2610i
0.0710 - 0.2240i
1.2730 + 0.0000i
0.0810 - 0.9220i
0.4960 - 1.2610i
0.0810 + 0.9220i
2.5520 + 0.0000i
3.9670 + 0.4240i
0.0260 - 0.7560i
1.5730 + 1.4700i
0.9180 + 0.5760i
-0.1040 + 0.3540i
0.3120 + 0.4890i
6.8090 + 0.0000i
0.0710 + 0.2240i
0.4960 + 1.2610i
0.6980 + 0.5220i
0.9180 - 0.5760i
3.9670 - 0.4240i
-0.1040 - 0.3540i
0.0260 + 0.7560i
0.3120 - 0.4890i
1.5730 - 1.4700i
0.6980 - 0.5220i
8.5130 + 0.0000i
is equal to W* X W
1.0e+03 *
0.6080 + 0.0000i
0.6980 + 0.5220i
0.9180 - 0.5760i
12
3.9670 - 0.4240i
-0.1040 - 0.3540i
0.0710 - 0.2240i
1.2730 + 0.0000i
0.0810 - 0.9220i
0.4960 - 1.2610i
0.0810 + 0.9220i
2.5520 + 0.0000i
3.9670 + 0.4240i
0.0260 - 0.7560i
1.5730 + 1.4700i
0.0260 + 0.7560i
0.3120 - 0.4890i
1.5730 - 1.4700i
0.6980 - 0.5220i
8.5130 + 0.0000i
Example 2:
To check if the given operator is normal or not
Enter the operator
[1 -1 i -i;3i 4 2i 8;1 3i 7 4i;1 -i i -1]
The operator is not normal
for W X W*
Columns 1 through 3
4.0000 + 0.0000i
-2.0000 -11.0000i
-3.0000 +10.0000i
-2.0000 +11.0000i
93.0000 + 0.0000i
0.0000 -27.0000i
-3.0000 -10.0000i
0.0000 +27.0000i
75.0000 + 0.0000i
2.0000 + 0.0000i
-6.0000 - 7.0000i
-2.0000 +11.0000i
Column 4
2.0000 + 0.0000i
-6.0000 + 7.0000i
-2.0000 -11.0000i
13
4.0000 + 0.0000i
is not equal to W* X W
Columns 1 through 3
12.0000 + 0.0000i
-1.0000 -10.0000i
13.0000 + 2.0000i
-1.0000 +10.0000i
27.0000 + 0.0000i
-1.0000 -14.0000i
13.0000 - 2.0000i
-1.0000 +14.0000i
55.0000 + 0.0000i
-1.0000 +21.0000i
44.0000 + 0.0000i
-1.0000 -13.0000i
Column 4
-1.0000 -21.0000i
44.0000 + 0.0000i
-1.0000 +13.0000i
82.0000 + 0.0000i
14
3.3
Quasinormal operators
Aim: To Create a program that shows if a given operator is Quasinormal or not.
Algorithm
1. Read the operator from the user and store it in the variable w.
2. Convert the operator w to its conjugate w’ and store it in the variable wconj.
3. Multiply the operator w’ with w and store the result in the operator wconjw.
4. Multiply the operator wconjw and w to obtain the value for (W*w)W.
5. Multiply the operator w and wconjw to obtain the value for W(W*w).
6. Check if the operator w*wconjw is equal to the operator wconjw*w.
7. If the operator w*wconjw is equal to the operator wconjw*w, then print the message
”for W(W*W), is equal to (W*W)W” and display the operators wconjw*w and
w*wconjw.
8. If the operator w*wconjw is not equal to the operator wconjw*w, then print the message ”for W(W*W), is not equal to (W*W)W” and display the operators wconjw*w
and w*wconjw.
Program
heading="To check if the given operator is quasinormal or not\n";
prompt="Enter the operator\n";
15
result1="The given operator is quasinormal\n";
result2="The given operator is not quasinormal\n";
fprintf(heading);
w=input(prompt);
wconj=w’;
wconjw=wconj*w;
if w*wconjw==wconjw*w
fprintf(result1);
disp("for W(W*W),");
disp(w*wconjw);
disp("is equal to (W*W)W");
disp(wconjw*w);
else
fprintf(result2);
disp("for W(W*W),");
disp(w*wconjw);
disp("is not equal to (W*W)W");
disp(wconjw*w);
end
Now let us run the program an operator to see whether the given operator is Quasinormal
or not.
16
Example 1:
To check if the given operator is quasinormal or not
Enter the operator
[6i 12i;12i 24i]
The given operator is quasinormal
for W(W*W),
1.0e+04 *
0.0000 + 0.5400i
0.0000 + 1.0800i
0.0000 + 1.0800i
0.0000 + 2.1600i
is equal to (W*W)W
1.0e+04 *
0.0000 + 0.5400i
0.0000 + 1.0800i
0.0000 + 1.0800i
0.0000 + 2.1600i
Example 2:
To check if the given operator is quasinormal or not
Enter the operator
[1 -1 i -i;3i 4 2i 8;1 3i 7 4i;1 -i i -1]
The given operator is not quasinormal
for W(W*W),
1.0e+02 *
17
0.3600 + 0.0400i
-0.4200 - 0.5500i
0.0100 + 0.7200i
4.6200 - 0.0500i
-0.1800 - 0.1100i
-0.0800 + 3.4500i
4.9200 - 0.0500i
-0.5900 - 0.3800i
0.0000 + 0.7100i
-0.5800 - 1.0400i
-0.0800 + 2.7000i
8.6900 - 0.0500i
-0.1100 - 0.2100i
-0.0800 + 5.3000i
0.2500 - 0.0700i
-0.9600 - 0.6600i
is not equal to (W*W)W
1.0e+02 *
0.5400 - 0.2200i
-0.4300 + 0.0000i
1.3200 + 0.2300i
1.5100 - 0.5700i
-0.1700 - 0.0100i
-0.0400 + 2.2400i
3.4600 + 0.1000i
2.1600 - 1.0600i
-0.2800 + 0.7800i
-0.1500 - 0.1900i
0.4200 + 0.7700i
2.3800 - 0.0300i
0.2500 + 0.0800i
-0.0900 + 3.0600i
0.8000 + 1.4000i
3.4300 - 0.0300i
18
3.4
Quasi P normal operators
Aim: To Create a program that shows if a given operator is Quasi P normal or not.
Algorithm
1. Begin
2. Read the value of the operator from the user using input(prompt)
3. Convert the operator to its conjugate using wconj=w’;
4. Concatenate the operator and its conjugate to form the product W*W(W+W*)
and store the value in wconjw*wpwconj.
5. Do the same for (W+W*)W*W and store the value in wpwconj*wconjw.
6. Convert the product to its complex conjugate using wconjw=wconj*w
7. Check if the complex conjugate of the product is equal to the product of the complex
conjugate and check the condition if wconjw*wpwconj==wpwconj*wconjw.
8. If the statement is true, then print the first result using fprintf(result1) and display
the value of wconjw*wpwconj and wpwconj*wconjw using disp function.
9. If the statement is false, then print the second result using fprintf(result2) and
display the value of wpwconj*wconjw and wconjw*wpwconj using disp function.
10. End
19
Program
heading="To check if the given operator is quasi p normal or not\n";
prompt="Enter the operator\n";
result1="The given operator is quasi p normal\n";
result2="The given operator is not quasi p normal\n";
fprintf(heading);
w=input(prompt);
wconj=conj(w)’;
wpwconj=w+wconj;
wconjw=wconj*w;
if wconjw*wpwconj==wpwconj*wconjw
fprintf(result1);
disp("for W*W(W+W*),");
disp(wconjw*wpwconj);
disp("is equal to (W+W*)W*w");
disp(wpwconj*wconjw);
else
fprintf(result2);
disp("for W*W(W+W*),");
disp(wconjw*wpwconj);
disp("is not equal to (W+W*)W*W");
disp(wpwconj*wconjw);
end
20
Now let us run the program an operator to see whether the given operator is Quasi P
normal or not.
Example 1:
To check if the given operator is quasi p normal or not
Enter the operator
[6i 12i;12i 24i]
The given operator is quasi p normal
for W*W(W+W*),
1.0e+04 *
0.0000 - 1.0800i
0.0000 - 2.1600i
0.0000 - 2.1600i
0.0000 - 4.3200i
is equal to (W+W*)W*w
1.0e+04 *
0.0000 - 1.0800i
0.0000 - 2.1600i
0.0000 - 2.1600i
0.0000 - 4.3200i
21
3.5
n power normal operators
Aim: To Create a program that shows if a given operator is n power normal or not.
Algorithm
1. 1. Begin
2. Read the starting range for n, and ending range for n from the user using fprintf(prompte)
and fprintf(promptn)
3. Convert the operator to its conjugate using wconj=w’
4. Start a for loop and set the value of i from e to n for the range of n.
5. Inside the for loop Calculate wn = wi
6. Check if wnwconj is equal to wconj*wn for each i.
7. If the statement is true, then print the first result using fprintf(result1) and display
the value of wn*wconj and wconj*wn using disp function and and terminate the
loop using break statement.
8. If the statement is false, then increase the value of i by 1 and repeat steps 5-7 until
the statement is true or the ending range for n is reached
9. End
Program
heading="To check if the given operator is n power normal or not\n";
22
prompt="Enter the operator\n";
prompte="Enter a starting range for n\n";
promptn="Enter an ending range for n\n";
result1="The operator is n power normal for n= %d\n";
result2="The operator is not n power normal for n=%d\n";
fprintf(heading);
w=input(prompt);
e=input(prompte);
n=input(promptn);
wconj=w’;
for i=e:n
wn=w^i;
if wn*wconj==wconj*wn
fprintf(result1,i);
disp("for W^n X W*,");
disp(wn*wconj);
disp("is equal to W* X W^n");
disp(wconj*wn);
break;
else
fprintf(result2,i);
end
end
23
Now let us run the program for an operator to see whether the given operator is n power
normal or not. Example 1:
To check if the given operator is n power normal or not
Enter the operator
[6i 12i;12i 24i]
Enter a starting range for n
2
Enter an ending range for n
10
The operator is n power normal for n= 2
for W^n X W*,
1.0e+04 *
0.0000 + 0.5400i
0.0000 + 1.0800i
0.0000 + 1.0800i
0.0000 + 2.1600i
is equal to W* X W^n
1.0e+04 *
0.0000 + 0.5400i
0.0000 + 1.0800i
0.0000 + 1.0800i
0.0000 + 2.1600i
24
3.6
K-Quasinormal operators
Aim: To Create a program that shows if a given operator is K-Quasinormal or not.
Algorithm
1. Begin.
2. Read the operator start and end for range of k from the user using the fprintf
function and store it in the variables e and k respectively
3. Convert the operator to its conjugate using wconj=w’.
4. Calculate and store the value of (W* X W) in the variable wconjw.
5. Create a ”for” loop that runs from the start of range of k to the end for range of k.
6. Calculate and store the value of w ∗ wconjwi in the variable wk inside the for loop.
7. Check if wk*w == w*wk.
8. If the statement is true, then print the first result using fprintf(result1) and display
the value of wk*w and w*wk using disp function.
9. If the statement is false, then increase the value of i by 1 and repeat steps 5-7 until
the statement is true or the end for range of k is reached.
10. End.
Program
25
heading="To check if the given operator is k quasi normal or not\n";
prompt="Enter the operator\n";
promptn="Enter an end for range of k\n";
prompte="Enter a start for range of k\n"
result1="The operator is k quasi normal for k= %d\n";
result2="The operator is not k quasi normal for k=%d\n";
fprintf(heading);
w=input(prompt);
e=input(prompte);
k=input(promptn);
wconj=w’;
wconjw=wconj*w;
for i=e:k
wk=wconjw^i;
if wk*w==w*wk
fprintf(result1,i);
disp("for");
disp(wk*w);
disp("equal to ");
disp(w*wk);
break;
26
else
fprintf(result2,i);
disp("for");
disp(wk*w);
disp("not equal to ");
disp(w*wk);
end
end
Now let us run the program for an operator to see whether the given operator is KQuasinormal or not. Example 1:
To check if the given operator is k quasi normal or not
Enter the operator
[1 i i;i 1 i;i i 1]
Enter a start for range of k
1
Enter an end for range of k
10
The operator is k quasi normal for k= 1
for
2601
3330
3330
3330
2601
3330
3330
3330
2601
equal to
27
2601
3330
3330
3330
2601
3330
3330
3330
2601
28
3.7
K-Quasi P normal operators
Algorithm
1. Begin
2. Read the operator from the user using the fprintf function.
3. Prompt the user to enter the start and end value for the range of k and store the
values in the variables e and k respectively.
4. Convert the operator to its conjugate using wconj=w’.
5. Create a for loop that runs from e to k.
6. Calculate wk=w ∗ wconjwi inside the for loop for each i.
7. Check if wk*wpwconj == wpwconj*wk for each i.
8. If the statement is true, then print the first result using fprintf(result1) and display
the value of wk*wpwconj and wpwconj*wk using disp function and terminate the
loop by break statement.
9. If the statement is false, then increase the value of i by 1 and repeat steps 5-7 until
the statement is true or the range for k is reached
10. End
Program
heading="To check if the given operator is k quasi p normal or not\n";
prompt="Enter the operator\n";
29
prompte="Enter a start for the range of k\n";
promptn="Enter an end for the range for k\n";
result1="The operator is k quasi p normal for k= %d\n";
result2="The operator is not k quasi p normal for k=%d\n";
fprintf(heading);
w=input(prompt);
e=input(prompte);
k=input(promptn);
wconj=w’;
wconjw=wconj*w;
wpwconj=w+wconj;
for i=e:k
wk=wconjw^i;
if wk*wpwconj==wpwconj*wk
fprintf(result1,i);
disp("for");
disp(wk*wpwconj);
disp("equal to ");
disp(wpwconj*wk);
break;
else
fprintf(result2,i);
30
end
end
Now let us run the program for an operator to see whether the given operator is K-Quasi
P normal or not. Example 1:
To check if the given operator is k quasi p normal or not
Enter the operator
[1 i i;i 1 i;i i 1]
Enter a start for the range of k
10
Enter an end for the range for k
20
The operator is k quasi p normal for k= 10
for
1.0e+09 *
6.9736
6.9736
6.9736
6.9736
6.9736
6.9736
6.9736
6.9736
6.9736
6.9736
6.9736
6.9736
6.9736
6.9736
6.9736
6.9736
6.9736
6.9736
equal to
1.0e+09 *
31
3.8
n power Quasinormal operators
Aim: To Create a program that shows if a given operator is n power Quasinormal or
not.
Algorithm
1. Begin
2. Read the operator from the user.
3. Read the start and end value for range of n from the user using fprintf function and
store the values in the variables e and k.
4. Convert the operator to its conjugate using wconj=w’
5. Create a loop that runs from the start value for range of n (i.e., ’e’) to the end value
for range of n (i.e.,’k’)
6. Calculate wn = wi for each i
7. Check if wn*wconjw is equal to wconjw*wn
8. If the statement is true, then print the first result using fprintf(result1) and display
the value of wn*wconjw and wconjw*wn using disp function.
9. If the statement is false, then increase the value of i by 1 and repeat steps 5-7 until
the statement is true or the end value for range of n is reached
10. End
32
Program
heading="To check if the given operator is n power quasinormal or not";
prompt="Enter the operator";
prompte="Enter a start value for range for n";
promptn="Enter an end value for range of n"
result1="The operator is n power quasinormal for n= %d\n";
result2="The operator is not n power quasinormal for n=%d\n";
fprintf(heading);
w=input(prompt);
e=input(prompte);
n=input(promptn);
wconj=w’;
wconjw=wconj*w;
for i=1:n
wn=w^i;
if wn*wconjw==wconjw*wn
fprintf(result1,i);
disp(for W^n(W*W),);
disp(wn*wconjw);
disp(is equal to (W*W)W^n);
disp(Wconjw*wn);
break;
else
33
fprintf(result2,i);
end
end
Now let us run the program an operator to see whether the given operator is n power
Quasinormal.
Example 1:
To check if the given operator is n power quasinormal or not
Enter the operator
[1 i i i;i 1 i i;i i 1 i;i i i 1]
Columns 1 through 3
1.0000 + 0.0000i
0.0000 + 1.0000i
0.0000 + 1.0000i
0.0000 + 1.0000i
1.0000 + 0.0000i
0.0000 + 1.0000i
0.0000 + 1.0000i
0.0000 + 1.0000i
1.0000 + 0.0000i
0.0000 + 1.0000i
0.0000 + 1.0000i
0.0000 + 1.0000i
Column 4
0.0000 + 1.0000i
0.0000 + 1.0000i
0.0000 + 1.0000i
1.0000 + 0.0000i
Enter a start value for range for n
34
1
Enter an end value for range of n
10
The operator is n power quasinormal for n= 1
for W^n(W*W),
Columns 1 through 3
4.0000 + 6.0000i
2.0000 + 8.0000i
2.0000 + 8.0000i
2.0000 + 8.0000i
4.0000 + 6.0000i
2.0000 + 8.0000i
2.0000 + 8.0000i
2.0000 + 8.0000i
4.0000 + 6.0000i
2.0000 + 8.0000i
2.0000 + 8.0000i
2.0000 + 8.0000i
4.0000 + 6.0000i
2.0000 + 8.0000i
2.0000 + 8.0000i
2.0000 + 8.0000i
4.0000 + 6.0000i
2.0000 + 8.0000i
2.0000 + 8.0000i
2.0000 + 8.0000i
4.0000 + 6.0000i
Column 4
2.0000 + 8.0000i
2.0000 + 8.0000i
2.0000 + 8.0000i
4.0000 + 6.0000i
is equal to (W*W)W^n
Columns 1 through 3
35
2.0000 + 8.0000i
2.0000 + 8.0000i
Column 4
2.0000 + 8.0000i
2.0000 + 8.0000i
2.0000 + 8.0000i
4.0000 + 6.0000i
36
2.0000 + 8.0000i
3.9
n power K-Quasinormal operators
Aim: To Create a program that shows if a given operator is K-Quasinormal or not.
Algorithm
1. 1. Begin
2. Read the operator, start value for range of n, end value for range of n and store it
in the variables e and n.
3. Read the start value and end value for range of k from the user using fprintf function
and store the values in the variables e1 and k
4. Convert the operator to its conjugate using wconj=w’
5. Create a loop that runs from the start value for range of n to the end value for
range of n
6. Create a second loop that runs from the start value for range of k to the end value
for range of k
7. Inside the first loop, calculate wn = wi
8. Inside the second loop, calculate wk = wj
9. Inside both loops, check if wn*wconjwk is equivalent to wconjwk*wn
10. If the statement is true, then print the first result using fprintf(result1) and display
the value of wn*wconjwk and wconjwk*wn using disp function.
37
11. If the statement is false, then increase the value of i by 1 and the value of j by 1
and repeat steps 5-11 until the statement is true or the end value for range of n is
reached
12. After the first loop, increase the value of i by 1 and repeat steps 5-11 until the end
value for range of k is reached or until the statement is true.
13. End
Program
heading="To check if the given operator is n power k quasinormal or not";
prompt="Enter the operator";
prompte="Enter a start for range for n";
promptn="Enter an end for range for n";
prompte1="Enter a start for range for k";
promptn1="Enter an end for range for k";
result1="The operator is n power k quasinormal for n= %d and k= %d\n";
result2="The operator is not n power k quasinormal for n=%d and k= %d\n";
fprintf(heading);
w=input(prompt);
e=input(prompte);
n=input(promptn);
e1=input(prompte1);
k=input(promptn1);
wconj=w’;
wconjw=wconj*w;
38
for i=e:n
wn=w^n;
for j=e1:k
wconjwk=wconjw^k;
if wn*wconjwk==wconjwk*wn
fprintf(result1,i,j);
disp("for (W^n)*(W*W)^k,");
disp(wn*wconjwk);
disp("is equal to (W*W)^k(W^n)");
disp(wconjwk*wn);
break;
else
fprintf(result2,i,j);
end
end
end
Now let us run the program an operator to see whether the given operator is n power
K-Quasinormal or not.
39
Example 1:
To check if the given operator is n power k quasinormal or not
Enter the operator
[2 4 8 16;4 8 16 32;8 16 32 64;16 32 64 128]
Enter a start for range for n
1
Enter an end for range for n
5
Enter a start for range for k
1
Enter an end for range for k
5
The operator is n power k quasinormal for n= 5 and k= 1
for (W^n)*(W*W)^k,
1.0e+33 *
0.0337
0.0674
0.1347
0.2694
0.0674
0.1347
0.2694
0.5388
0.1347
0.2694
0.5388
1.0776
0.2694
0.5388
1.0776
2.1552
is equal to (W*W)^k(W^n)
1.0e+33 *
0.0337
0.0674
0.1347
0.2694
40
0.0674
0.1347
0.2694
0.5388
0.1347
0.2694
0.5388
1.0776
0.2694
0.5388
1.0776
2.1552
41
3.10
n power K-Quasi P normal operators
Aim: To Create a program that shows if a given operator is n power K-Quasi P normal
or not.
Algorithm
1. Begin
2. Prompt the user to enter the operator using fprintf funtion
3. Prompt the user to enter the start and end for range of n using input function and
store the values in e and n variables
4. Prompt the user to enter the end for range of k using input function and store the
values in the variables e1 and k
5. Convert the operator to its conjugate using wconj=w’
6. Create a loop that runs from the start for range of n to the end for range of n
7. Create a second loop inside the first loop that runs from the start for range of k to
the end for range of k
8. Inside the first loop, calculate wn = wi
9. Inside the second loop, calculate wk = wj
10. Calculate (W n + W ∗n ) and store it in the variable wnpwconjn
11. Calculate (W ∗ W )k and store it in the variable wconjwk
12. Inside both loops, check if wnpwconjn*wconjwk is equal to wconjwk*wnpwconjn
42
13. If the statement is true, then print the first result using fprintf(result1) and display
the value of wn*wconjwk and wconjwk*wn using disp funciton and break the loops.
14. If the statement is false, then increase the value of i by 1 and the value of j by 1
and repeat steps 11-12 until the statement is true or the end value for range of n is
reached
15. After the first loop, increase the value of i by 1 and repeat steps 8-15 until the end
value for range of k is reached
16. End
Program
heading="To check if the given operator is
n power k quasi p normal or not";
prompt="Enter the operator";
prompte="Enter a start for range of n";
promptn="Enter an end for range of n";
prompte1="Enter a start for range of k";
promptk="Enter an end for range of k";
result1="The operator is n power k quasi p normal for n= %d and k= %d\n";
result2="The operator is not n power k quasi p normal for n=%d and k= %d\n";
fprintf(heading);
w=input(prompt);
e=input(prompte);
n=input(promptn);
43
e1=input(prompte1);
k=input(promptk);
wconj=conj(w)’;
wconjw=wconj*w;
for i=e:n
wn=w^n;
wconjn=wconj^n;
wnpwconjn=wn+wconjn;
for j=e1:k
wconjwk=wconjw^k;
if wnpwconjn*wconjwk==wconjwk*wnpwconjn
fprintf(result1,i,j);
disp("for (W^n)*(W*W)^k,");
disp(wn*wconjwk);
disp("is equal to (W*W)^k(W^n)");
disp(wconjwk*wn);
break;
else
fprintf(result2,i,j);
end
end
end
Now let us run the program an operator to see whether the given operator is n power
44
K-Quasi P normal. Example 1:
To check if the given operator is n power k quasi p normal or not
Enter the operator
[1 i i;i 1 i;i i 1]
Enter a start for range of n
1
Enter an end for range of n
5
Enter a start for range of k
1
Enter an end for range of k
5
The operator is n power k quasi p normal for n= 4 and k= 1
for (W^n)*(W*W)^k,
1.0e+04 *
-2.5481 + 0.5460i
-2.5481 + 0.5332i
-2.5481 + 0.5332i
-2.5481 + 0.5332i
-2.5481 + 0.5460i
-2.5481 + 0.5332i
-2.5481 + 0.5332i
-2.5481 + 0.5332i
-2.5481 + 0.5460i
is equal to (W*W)^k(W^n)
1.0e+04 *
-2.5481 + 0.5460i
-2.5481 + 0.5332i
-2.5481 + 0.5332i
-2.5481 + 0.5332i
-2.5481 + 0.5460i
-2.5481 + 0.5332i
45
-2.5481 + 0.5332i
-2.5481 + 0.5332i
46
-2.5481 + 0.5460i
4
Programs II
4.1
r+ic type operators
This section deals with operators entries r+ic, for (r,c) are row and column indices of
the matrix. We write a Matlab program that describe the properties of these type of
operators of different sizes.
Program
disp("(r+ic) matrix");
prompt1="size of the matrix";
n=input(prompt1);
i=sqrt(-1);
t=eye(n);
I=eye(n);
for r=1:n
for c=1:n
t(r,c)=(r+i*c);
end
end
disp("The given operator:");
disp(t);
tstar=t’;
disp("conjugate of t:");
47
disp(tstar);
disp("norm of t:");
nt=norm(t);
disp(nt);
tsq=t^2;
disp("t square:");
disp(tsq);
ntsq=norm(tsq);
disp("norm of t squared:");
disp(ntsq);
nsqtsq=(ntsq)^2;
disp("square of norm t squared:");
disp(nsqtsq);
tq=t^3;
disp("t cube:");
disp(tq);
disp("norm of t cube:");
ntq=norm(tq);
disp(ntq);
ntqnt=ntq*nt;
disp("norm of t cube * norm of t:");
disp(ntqnt);
%%checking if the operator is normal, self adjoint and unitary
if(t*tstar==tstar*t)
disp("The given operator is normal");
else
48
disp("The given operator is not normal");
end
if(t==tstar)
disp("The given operator is self adjoint");
else
disp("The given operator is not self adjoint");
end
if(t*tstar==I)
disp("The given operator is unitary");
else
disp("The given operator is not unitary");
end
%%checking if the operator is an isometry
if(t*tstar==I)
disp("The given operator is an isometry");
else
disp("The given operator is not an isometry");
end
%%checking if the operator is quasinormal
if(t*(tstar*t)==(tstar*t)*t)
disp("The given operator is quasinormal");
else
disp("The given operator is not quasinormal");
end
49
%%checking if the operator is m quasi paranormal for a given range
mqpn="Checking if the operator is m quasi paranormal, \nEnter m";
a=input(mqpn);
for m=1:a
if(nsqtsq<=m*ntqnt)
result=’when m is %d, operator is m quasi paranormal\n
fprintf(result,m,nsqtsq,m*ntqnt);
break;
else
nresult=’when m is %d, operator is not m quasi paranormal\n
fprintf(nresult,m,nsqtsq,m*ntqnt);
end
end
Now let us run different outputs for different size of (r+ic) type operators.
50
Example 1:
(r+ic) matrix
size of the matrix
3
The given operator:
1.0000 + 1.0000i
1.0000 + 2.0000i
1.0000 + 3.0000i
2.0000 + 1.0000i
2.0000 + 2.0000i
2.0000 + 3.0000i
3.0000 + 1.0000i
3.0000 + 2.0000i
3.0000 + 3.0000i
1.0000 - 1.0000i
2.0000 - 1.0000i
3.0000 - 1.0000i
1.0000 - 2.0000i
2.0000 - 2.0000i
3.0000 - 2.0000i
1.0000 - 3.0000i
2.0000 - 3.0000i
3.0000 - 3.0000i
conjugate of t:
norm of t:
9.1416
t square:
0.0000 +17.0000i
-6.0000 +20.0000i -12.0000 +23.0000i
6.0000 +20.0000i
0.0000 +26.0000i
-6.0000 +32.0000i
12.0000 +23.0000i
6.0000 +32.0000i
0.0000 +41.0000i
norm of t squared:
83.5692
51
square of norm t squared:
6.9838e+03
t cube:
1.0e+02 *
-1.0800 + 1.0800i
-1.6800 + 0.9000i
-2.2800 + 0.7200i
-0.9000 + 1.6800i
-1.6800 + 1.6800i
-2.4600 + 1.6800i
-0.7200 + 2.2800i
-1.6800 + 2.4600i
-2.6400 + 2.6400i
norm of t cube:
763.9581
norm of t cube * norm of t:
6.9838e+03
The given operator is normal
The given operator is not self adjoint
The given operator is not unitary
The given operator is not an isometry
The given operator is quasinormal
Checking if the operator is m quasi paranormal,
Enter m
5
when m is 1, operator is m quasi paranormal
for 6.983814e+03 is less than or equal to 6.983814e+03
52
Example 2:
(r+ic) matrix
size of the matrix
6
The given operator:
Columns 1 through 3
1.0000 + 1.0000i
1.0000 + 2.0000i
1.0000 + 3.0000i
2.0000 + 1.0000i
2.0000 + 2.0000i
2.0000 + 3.0000i
3.0000 + 1.0000i
3.0000 + 2.0000i
3.0000 + 3.0000i
4.0000 + 1.0000i
4.0000 + 2.0000i
4.0000 + 3.0000i
5.0000 + 1.0000i
5.0000 + 2.0000i
5.0000 + 3.0000i
6.0000 + 1.0000i
6.0000 + 2.0000i
6.0000 + 3.0000i
1.0000 + 4.0000i
1.0000 + 5.0000i
1.0000 + 6.0000i
2.0000 + 4.0000i
2.0000 + 5.0000i
2.0000 + 6.0000i
3.0000 + 4.0000i
3.0000 + 5.0000i
3.0000 + 6.0000i
4.0000 + 4.0000i
4.0000 + 5.0000i
4.0000 + 6.0000i
5.0000 + 4.0000i
5.0000 + 5.0000i
5.0000 + 6.0000i
6.0000 + 4.0000i
6.0000 + 5.0000i
6.0000 + 6.0000i
Columns 4 through 6
conjugate of t:
53
Columns 1 through 3
1.0000 - 1.0000i
2.0000 - 1.0000i
3.0000 - 1.0000i
1.0000 - 2.0000i
2.0000 - 2.0000i
3.0000 - 2.0000i
1.0000 - 3.0000i
2.0000 - 3.0000i
3.0000 - 3.0000i
1.0000 - 4.0000i
2.0000 - 4.0000i
3.0000 - 4.0000i
1.0000 - 5.0000i
2.0000 - 5.0000i
3.0000 - 5.0000i
1.0000 - 6.0000i
2.0000 - 6.0000i
3.0000 - 6.0000i
4.0000 - 1.0000i
5.0000 - 1.0000i
6.0000 - 1.0000i
4.0000 - 2.0000i
5.0000 - 2.0000i
6.0000 - 2.0000i
4.0000 - 3.0000i
5.0000 - 3.0000i
6.0000 - 3.0000i
4.0000 - 4.0000i
5.0000 - 4.0000i
6.0000 - 4.0000i
4.0000 - 5.0000i
5.0000 - 5.0000i
6.0000 - 5.0000i
4.0000 - 6.0000i
5.0000 - 6.0000i
6.0000 - 6.0000i
Columns 4 through 6
norm of t:
32.8909
t square:
1.0e+02 *
Columns 1 through 3
54
0.0000 + 0.9700i
-0.2100 + 1.0300i
-0.4200 + 1.0900i
0.2100 + 1.0300i
0.0000 + 1.1500i
-0.2100 + 1.2700i
0.4200 + 1.0900i
0.2100 + 1.2700i
0.0000 + 1.4500i
0.6300 + 1.1500i
0.4200 + 1.3900i
0.2100 + 1.6300i
0.8400 + 1.2100i
0.6300 + 1.5100i
0.4200 + 1.8100i
1.0500 + 1.2700i
0.8400 + 1.6300i
0.6300 + 1.9900i
-0.6300 + 1.1500i
-0.8400 + 1.2100i
-1.0500 + 1.2700i
-0.4200 + 1.3900i
-0.6300 + 1.5100i
-0.8400 + 1.6300i
-0.2100 + 1.6300i
-0.4200 + 1.8100i
-0.6300 + 1.9900i
0.0000 + 1.8700i
-0.2100 + 2.1100i
-0.4200 + 2.3500i
0.2100 + 2.1100i
0.0000 + 2.4100i
-0.2100 + 2.7100i
0.4200 + 2.3500i
0.2100 + 2.7100i
0.0000 + 3.0700i
Columns 4 through 6
norm of t squared:
1.0818e+03
square of norm t squared:
1.1703e+06
t cube:
1.0e+03 *
Columns 1 through 3
55
-2.1420 + 2.1420i
-2.8140 + 1.8270i
-3.4860 + 1.5120i
-1.8270 + 2.8140i
-2.6250 + 2.6250i
-3.4230 + 2.4360i
-1.5120 + 3.4860i
-2.4360 + 3.4230i
-3.3600 + 3.3600i
-1.1970 + 4.1580i
-2.2470 + 4.2210i
-3.2970 + 4.2840i
-0.8820 + 4.8300i
-2.0580 + 5.0190i
-3.2340 + 5.2080i
-0.5670 + 5.5020i
-1.8690 + 5.8170i
-3.1710 + 6.1320i
-4.1580 + 1.1970i
-4.8300 + 0.8820i
-5.5020 + 0.5670i
-4.2210 + 2.2470i
-5.0190 + 2.0580i
-5.8170 + 1.8690i
-4.2840 + 3.2970i
-5.2080 + 3.2340i
-6.1320 + 3.1710i
-4.3470 + 4.3470i
-5.3970 + 4.4100i
-6.4470 + 4.4730i
-4.4100 + 5.3970i
-5.5860 + 5.5860i
-6.7620 + 5.7750i
-4.4730 + 6.4470i
-5.7750 + 6.7620i
-7.0770 + 7.0770i
Columns 4 through 6
norm of t cube:
3.5582e+04
norm of t cube * norm of t:
1.1703e+06
The given operator is normal
The given operator is not self adjoint
The given operator is not unitary
56
The given operator is not an isometry
The given operator is quasinormal
Checking if the operator is m quasi paranormal,
Enter m
5
when m is 1, operator is not m quasi paranormal
for 1.170310e+06 is not less than or equal to 1.170310e+06
when m is 2, operator is m quasi paranormal
for 1.170310e+06 is less than or equal to 2.340620e+06
57
4.2
2r*3ic operators
This section deals with operators entries 2r*3ic, for (r,c) are row and column indices of
the matrix. We write a Matlab program that describe the properties of these type of
operators of different sizes.
The program is the same as the previous one with little changes for creating the operator. In the above program a set of code is replaced with new code that produces 2r*3c
operators.
Program
The following Matlab code of the previous program
disp("(r+ic) matrix");
prompt1="size of the matrix";
n=input(prompt1);
i=sqrt(-1);
t=eye(n);
I=eye(n);
for r=1:n
for c=1:n
t(r,c)=(r+i*c);
end
end
is replaced with
58
disp("(2r*3ic) matrix");
prompt1="size of the matrix";
n=input(prompt1);
i=sqrt(-1);
t=eye(n);
I=eye(n);
for r=1:n
for c=1:n
t(r,c)=(2*r*3*c*i);
end
end
This new program produces (2r*3ic) type operators and returns its properties.
Now let us run different outputs for different size of (2r*3ic) type operators.
59
Example 1:
(2r*3ic) matrix
size of the matrix
3
The given operator:
0.0000 + 6.0000i
0.0000 +12.0000i
0.0000 +18.0000i
0.0000 +12.0000i
0.0000 +24.0000i
0.0000 +36.0000i
0.0000 +18.0000i
0.0000 +36.0000i
0.0000 +54.0000i
0.0000 - 6.0000i
0.0000 -12.0000i
0.0000 -18.0000i
0.0000 -12.0000i
0.0000 -24.0000i
0.0000 -36.0000i
0.0000 -18.0000i
0.0000 -36.0000i
0.0000 -54.0000i
conjugate of t:
norm of t:
84.0000
t square:
-504
-1008
-1512
-1008
-2016
-3024
-1512
-3024
-4536
norm of t squared:
7056
60
square of norm t squared:
49787136
t cube:
1.0e+05 *
0.0000 - 0.4234i
0.0000 - 0.8467i
0.0000 - 1.2701i
0.0000 - 0.8467i
0.0000 - 1.6934i
0.0000 - 2.5402i
0.0000 - 1.2701i
0.0000 - 2.5402i
0.0000 - 3.8102i
norm of t cube:
5.9270e+05
norm of t cube * norm of t:
4.9787e+07
The given operator is normal
The given operator is not self adjoint
The given operator is not unitary
The given operator is not an isometry
The given operator is quasinormal
Checking if the operator is m quasi paranormal,
Enter m
5
when m is 1, operator is not m quasi paranormal
for 49787136 is not less than or equal to 4.978714e+07
when m is 2, operator is m quasi paranormal
61
for 49787136 is less than or equal to 9.957427e+07
Example 2:
(2r*3ic) matrix
size of the matrix
4
The given operator:
Columns 1 through 2
0.0000 + 6.0000i
0.0000 +12.0000i
0.0000 +12.0000i
0.0000 +24.0000i
0.0000 +18.0000i
0.0000 +36.0000i
0.0000 +24.0000i
0.0000 +48.0000i
Columns 3 through 4
0.0000 +18.0000i
0.0000 +24.0000i
0.0000 +36.0000i
0.0000 +48.0000i
0.0000 +54.0000i
0.0000 +72.0000i
0.0000 +72.0000i
0.0000 +96.0000i
conjugate of t:
Columns 1 through 2
0.0000 - 6.0000i
0.0000 -12.0000i
0.0000 -12.0000i
0.0000 -24.0000i
62
0.0000 -18.0000i
0.0000 -36.0000i
0.0000 -24.0000i
0.0000 -48.0000i
Columns 3 through 4
0.0000 -18.0000i
0.0000 -24.0000i
0.0000 -36.0000i
0.0000 -48.0000i
0.0000 -54.0000i
0.0000 -72.0000i
0.0000 -72.0000i
0.0000 -96.0000i
norm of t:
180
t square:
-1080
-2160
-3240
-4320
-2160
-4320
-6480
-8640
-3240
-6480
-9720
-12960
-4320
-8640
-12960
-17280
norm of t squared:
32400
square of norm t squared:
1.0498e+09
t cube:
63
1.0e+06 *
Columns 1 through 2
0.0000 - 0.1944i
0.0000 - 0.3888i
0.0000 - 0.3888i
0.0000 - 0.7776i
0.0000 - 0.5832i
0.0000 - 1.1664i
0.0000 - 0.7776i
0.0000 - 1.5552i
Columns 3 through 4
0.0000 - 0.5832i
0.0000 - 0.7776i
0.0000 - 1.1664i
0.0000 - 1.5552i
0.0000 - 1.7496i
0.0000 - 2.3328i
0.0000 - 2.3328i
0.0000 - 3.1104i
norm of t cube:
5832000
norm of t cube * norm of t:
1.0498e+09
The given operator is normal
The given operator is not self adjoint
The given operator is not unitary
The given operator is not an isometry
64
The given operator is quasinormal
Checking if the operator is m quasi paranormal,
Enter m
5
when m is 1, operator is m quasi paranormal
for 1049760000 is less than or equal to 1049760000
65
Chapter 5
Conclusion
On a complex Hilbert space, we have examined numerous properties of various limited
linear operators in this project. We were able to rapidly check these features for each given
operator from a complex Hilbert space by using Matlab. Making functions in Matlab for
each property reduces the amount of effort and manual computations required while also
providing a clear grasp of these kinds of operators. This method can also be used to
create instances that disprove particular properties or theorems. With this the project is
completed.
66
Bibliography
[1] Dr.T. Veluchamy and Dr. SNS, k-Quasi-P-Normal Composition, Weighted
composition and composite multiplication operators on the complex hilbert space,
International journal of pure and applied mathematics, Vol. 119, No. 12, ISSN:
1314-3395, (2018).
[2] Valdete Rexhebeqaj Hamiti and Qefsere Doko Gjonbalaj, On M-quasi
paranormal operators, European Journal of Pure and Applied Mathematics, Vol. 15,
No. 3, 830-840, ISSN 1307-5543 (2022).
[3] Dr.T. Veluchamy and Dr. SNS, Power Classes of Composition Operators on
the Complex Hilbert Space and L2 space, International Journal of Pure and Applied
Mathematics, Vol. 118, No. 9, 137-161, ISSN 1311-8080(printed version), ISSN
1314-3395(on-line version)(2018)
[4] Ould Ahmed Mahmoud Sid Ahmed, On the class of n-power quasi normal operators on the hilbert space, Bull. Of Math.Anal., Vol 3,2, 213-228, (2008)
[5] D. Senthil Kumar, P. maheshwari Naik, R. santhi, k-Quasi - Norml operator, International Journal of Mathematics and Computation, Vol. 15,2, 99-105,
(2012)
67