New York Journal of Mathematics
New York J. Math. 5 (1999) 1{16.
On a Class of Toeplitz + Hankel Operators
Estelle L. Basor and Torsten Ehrhardt
Abstract. In this paper we study operators of the form M () = T ()+ H ()
where T () and H () are the Toeplitz and Hankel operators acting on `2 . We
investigate the connection between Fredholmness and invertibility of M ()
for functions 2 L1 (T). Using this relationship we establish necessary and
sucient conditions for the invertibility of M () with piecewise continuous .
Finally, we consider several stability problems related to M (), in particular
the stability of the nite section method.
Contents
1. Introduction
2. Basic properties of M ()
3. Fredholmness and invertibility for PC symbols
4. Stability of the nite section method for PC symbols
5. Stability for approximate identities of PC symbols
References
1.
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Introduction
This paper is devoted to the study of operators of the form
(1)
M () = T () + H ():
Here 2 L1 (T) is a Lebesgue measurable and essentially bounded function on the
unit circle T with Fourier coecients . The Toeplitz and Hankel operators are
dened as usual in terms of the innite matrices
; ;
(2)
H () = + +1 1 =0 :
T () = ; 1 =0 The operators are considered as acting on the Hilbert space `2 = `2 (Z+) of one-sided
square-summable sequences.
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Received May 12, 1998.
Mathematics Subject Classication. 47B35 secondary: 47A10, 47A35.
Key words and phrases. Toeplitz operators, Hankel operators, Fredholm Index, Invertibility,
Spectrum, Stability.
The rst author was supported in part by NSF Grant DMS-9623278.
The second author was supported in part by DAAD Grant 213/402/537/5.
1
c 1999 State University
of New York
ISSN 1076-9803/99
2
Estelle L. Basor and Torsten Ehrhardt
The operators M () represent a special case of Toeplitz + Hankel operators of
the form T () + H () with arbitrary functions and in L1 (T). Unfortunately,
it seem hopeless to ask for invertibility in this general setting, even if and are
continuous functions. However, in this paper we will show that the operators M ()
possess certain properties that make it possible to solve invertibility and related
problems for piecewise continuous .
In the next section we establish basic properties of M (). Among other things,
it turns out that in the case of even functions the invertibility of M () is almost
trivial. In the general case we obtain the remarkable result that M () is invertible
if and only if M () is Fredholm and has index zero. As in the Toeplitz case, this is
the key for studying invertibility.
In the third section we establish necessary and sucient conditions for the invertibility of M () in the case of piecewise continuous . These results are based
on Fredholm criteria that can be obtained from the work of Power 6] or Bottcher
and Silbermann 3]. Additional considerations are needed in order to determine the
Fredholm index. We also give a geometric description of the spectrum of M ().
As an application of the invertibility results, we investigate in the fourth section
the stability of the sequence of the nite sections
(3)
M () = T () + H ():
Here the n n Toeplitz and Hankel matrices are dened by
;
1:
; 1
(4)
H () = + +1 ;=0
T () = ; ;=0
The function is assumed to be piecewise continuous. The proof of the stability
criterion relies on results obtained by Roch and Silbermann 7] and Spitkovsky and
Tashbaev 8].
Another application is treated in the last section. There we examine the stability
problem for sequences fM ( )g as ! 1. Here f g are certain sequences of
(smooth) functions related to a given piecewise continuous function . For example,
the function may arise from any approximate identity of . The proofs are based
on results of 5].
The investigations taken up in this paper (in particular the afore-mentioned applications) are motivated by a recent paper 2]. There the asymptotic behavior of
the determinants of Toeplitz + Hankel matrices, det M () for certain piecewise
continuous functions is studied. The interest in the asymptotic behavior of these
determinants came from random matrix theory. For certain matrix ensembles one
needs to nd the asymptotic formula for the determinant of a sum of Wiener{Hopf
and Hankel integral operators, which is the continuous analogue of the problem
considered here. The desired asymptotic formula allows one to compute the distribution function for a discontinuous random variable. The discontinuous case is
particularly important because it corresponds to counting the number of eigenvalues of a random matrix in an interval. For more details on this topic we refer the
reader to the literature given in 2].
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2.
Basic properties of M ()
In this section we present basic properties of M () with symbols 2 L1(T).
We rst introduce some notation. Let `2 (Z) be the Hilbert space of two-sided
On a Class of Toeplitz
+ Hankel Operators
3
square-summable sequences. The Laurent operator acting on `2 (Z) with generating
function 2 L1 (T) is dened by the innite matrix
; (5)
L() = ; 1 =;1 :
It is well known that kL()k = kk1 . We also need the projection
0
(6)
P : (x ) 2Z 7! (y ) 2Z y = x0 ifif kk <
0
the associated projection Q = I ; P , and the ip operator
(7)
J : (x ) 2Z 7! (x;1; ) 2Z:
2
Note that JP = QJ , J = I , and that P , Q and J are selfadjoint. Moreover, we
have JL()J = L(e) where e is the function dened by e(t) = (1=t), t 2 T. It is
easy to observe that
(8)
T () = PL()P
H () = PL()JP
when identifying the image of P with the space `2 = `2(Z+). Hence
(9)
M () = PL()(I + J )P:
In the following, it will occasionally be convenient to identify `2 with the Hardy
space H 2 (T) and `2 (Z) with the Lebesgue space L2 (T) and to consider the operators
as acting on these spaces. In this setting, P is the Riesz projection, L() represents
the multiplication operator, and J is the operator that maps a function f (t) into
the function 1=t f (1=t).
As a rst result, we establish an estimate of the norm of M ().
p
Proposition 2.1. Let 2 L1(T). Then kk1 kM ()k 2 kk1.
Proof. Notice that
k(I + J )P k2 = kP (I + J ) (I + J )P k = 2 kP (I + J )P k = 2 kP k = 2:
p
Hence k(I + J )P k = 2. Taking (9) into account, the upper estimate follows.
Now let U denote the multiplication operator f (t) 7! t f (t) on L2 (T). Since
U U; = I and JU = U; J , it is easy to see that
U; M ()U = (U; PU )L()(U; PU ) + (U; PU; )L()J (U; PU ):
Because U; PU ! I and U; PU; ! 0 strongly on L2 (T) as n ! 1, we have
(10)
U; M ()U ! L()
2
strongly on L (T) as n ! 1. Note that U are isometries on L2 (T), and therefore
(10) implies the lower estimate.
The examples of functions (t) = 1 and (t) = t show that the lower and upper
estimates can in general not be improved.
Recall that for Toeplitz and Hankel operators the following relations hold:
(11)
T () = T ()T () + H ()H (e)
(12)
H () = T ()H () + H ()T (e):
Adding both equations, it follows that
M () = T ()M () + H ()M (e)
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Estelle L. Basor and Torsten Ehrhardt
and hence
(13)
M () = M ()M () + H ()M (e ; ):
This formula is in some sense the analogue of (11).
Let H 1 (T) be the Hardy space of all functions 2 L1 (T) for which = 0
for each n > 0. Moreover, denote by L1
even(T) the C -algebra of all functions
1
1
e
2 L (T) for which = . If 2 H (T) or 2 L1
even(T), then (13) simplies to
(14)
M () = M ()M ():
This means that there exist two classes of functions for which M () is multiplicative.
If 2 H 1 (T), then M () = T (). The properties of such Toeplitz operators are
well known. The second class, which consists of even functions, is considered next.
Theorem 2.2. The mapping : 7! M () is a -isomorphism from L1even(T)
onto a -subalgebra of L(`2 ).
Proof. Using the fact that JL() = L(e)J and J 2 = I , and equation (9), we can
write
M () = 1=2 P (I + J )L()(I + J )P:
Now it is easy to see that () = (), where denotes the complex conjugate of
. The fact that is multiplicative follows from (14). Because of Proposition 2.1,
the mapping has a trivial kernel. Hence is a -isomorphism.
We remark that this result implies in particular that kM ()k = kk1 for even
. This improvement of Proposition 2.1 can be proved
p directly by invoking the
above identity along with k(I + J )P k = kP (I + J )k = 2.
We see from Theorem 2.2 that the case of operators M () with even symbols is
especially simple. In particular, one can solve the invertibility problem completely.
In what follows, for a Banach algebra B , the notation GB stands for the group
of all invertible elements in this Banach algebra.
Corollary 2.3. Let 2 L1even(T). Then M () is invertible if and only if 2
GL1
even(T). Moreover, if this is true, then the inverse is given by M (;1 ).
In general, the invertibility problem is much more delicate. We will see below
how invertibility can be related to Fredholmness. A necessary Fredholm condition
is given next.
Proposition 2.4. Let 2 L1(T), and suppose that M () is Fredholm. Then
2 GL1 (T).
Proof. The proof is based on standard arguments. If M () is a Fredholm operator,
then there exist a > 0 and a (nite rank) projection K on the kernel of M ()
such that
kM ()f k 2 (T) + kKf k 2(T) kf k 2 (T)
for all f 2 H 2 (T). Putting Pf instead of f , this implies that
kM ()f k 2(T) + kPKPf k 2(T) + k(I ; P )f k 2 (T) kf k 2(T)
for all f 2 L2 (T). Replacing f by U f and observing again that U are isometries
on L2 (T), it follows that
kU; M ()U f k 2(T) + kPKPU f k 2(T) + kU; (I ; P )U f k 2(T) kf k 2(T):
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On a Class of Toeplitz
+ Hankel Operators
5
Now we take the limit n ! 1. Because U ! 0 weakly, we have PKPU ! 0
strongly. It remains to apply (10) and again the fact that U; PU ! I strongly.
We obtain
kL()f k 2(T) kf k 2(T):
This implies the desired assertion.
The next result is concerned with an assertion about the kernel and cokernel of
M (). It is the analogue of Coburn's result for Toeplitz operators. We begin with
the following observation. Given 2 L1 (T), dene
(15)
K = t 2 T (t) = (t;1 ) = 0 :
This denition of course depends on the choice of representatives for however
the following remarks are independent of that choice. Note that the characteristic
function is real and even. Thus M ( ) is a selfadjoint projection (Theorem 2.2).
Since M ()M ( ) = M ( ) = 0, we obtain that
(16)
im M ( ) ker M ():
If the set K has Lebesgue measure zero, then obviously M ( ) = 0, whereas if K
has a positive Lebesgue measure, then the image of M ( ) is innite dimensional.
The latter fact can be seen by decomposing K into pairwise disjoint and even sets
K1 : : : K with positive Lebesgue measure. Then M ( ) = M ( 1 ) + +
M ( n ), where M ( 1 ) : : : M ( n ) are mutually orthogonal projections which
are all nonzero (by Proposition 2.1). This shows that dim im M ( ) n for all n.
Theorem 2.5. Let 2 L1(T) and let K be as above. Then ker M () = im M ( )
or ker M () = f0g.
Proof. As before we consider the operator M () as being dened on H 2(T). Then
M () and its adjoint can be written in the form:
M () = PL()(I + J )P
M () = P (I + J )L()P:
Suppose that we have functions f+ g+ 2 H 2 (T) such that M ()f+ = 0 and
M ()g+ = 0 with g+ 6= 0. We have to show that f+ 2 im M ( ). Introducing the functions
f (t) = f+ (t) + t;1 f+ (t;1 )
f;(t) = (t)f (t)
g(t) = (t)g+ (t)
g;(t) = g(t) + t;1 g(t;1 )
it follows at once that f; g; 2 H;2 (T), where H;2 (T) denotes the kernel of P . We
obtain from the denition of g; that t;1 g; (t;1 ) = g; (t). Hence g; = 0 by checking
the Fourier coecients. It follows that t;1 g(t;1 ) = ;g(t). On the other hand, the
denition of f says that t;1 f (t;1 ) = f (t). This implies that (fg)(t;1 ) = ;(fg)(t).
Also from the above relations we conclude that fg = fg+ = f;g+ . Because
f; 2 H;2 (T) and g+ 2 H 2 (T), we have f;g+ 2 H;1 (T). Considering again the
Fourier coecients, this shows fg = f; g+ = 0. Since g+ 2 H 2 (T) and g+ 6= 0
the F. and M. Riesz Theorem says that g+ (t) 6= 0 almost everywhere on T, and
thus we obtain f; = 0. Hence f = 0. Because t;1 f (t;1 ) = f (t), we have also
ef = 0. The denition of the set K now implies that (1 ; )f = 0. Noting that
M (1 ; )f+ = 0, we nally arrive at f+ 2 im M ( ).
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6
Estelle L. Basor and Torsten Ehrhardt
Combining Theorem 2.5 with Proposition 2.4 we obtain the following result.
Corollary 2.6. Let 2 L1(T). If M () is Fredholm, then M () has a trivial
kernel or a trivial cokernel.
We remark that the previous assertion need not be true for Toeplitz + Hankel
operators in general. A counterexample is given by T () + H () with (t) = 1 and
(t) = ;t.
Another conclusion is the next statement, which reduces the invertibility problem
for M () to the Fredholm problem and the index computation.
Corollary 2.7. Let 2 L1(T). Then M () is invertible if and only if M () is
Fredholm and has index zero.
Note that in the case of a continuous function , the operator M () is just T ()
plus a compact operator. This in conjunction with the well known invertibility
result for Toeplitz operators puts us in position to obtain the following simple
result. If is continuous, then M () is invertible if and only if does not vanish
on T and has winding number zero.
Finally, we give a sucient invertibility condition that relies on factorization.
This condition follows directly from equation (14). Suppose that 2 L1 (T) can
be factored in the form = ; 0 with ; 2 GH 1 (T) and 0 2 GL1
even(T). Then
M () is invertible and the inverse equals
(17)
M ;1 () = M (;0 1 )T (;;1 ):
It seems very likely that one can formulate a necessary and sucient invertibility
(or Fredholm) criterion for M () in terms of a factorization of the symbol with
precise conditions on the factors as is well known in the Toeplitz case. However,
we must admit that we have not been able to accomplish such a result.
3.
Fredholmness and invertibility for P C symbols
Let PC stand for the C -algebra of all piecewise continuous functions on the unit
circle, i.e., functions for which the one-sided limits ( 0) := lim !+0 (e )
exist for each 2 T. In this section we want to establish Fredholmness and invertibility criteria for M () with piecewise continuous. With regard to Corollary 2.7,
it is necessary in this connection to determine the Fredholm index of M ().
The -subalgebra of L(`2 ) generated by Toeplitz and Hankel operators with
piecewise continuous symbols has been examined by Power 6]. The description
given there allows us to derive necessary and sucient conditions for Fredholmness
of operators of the form T () + H () with 2 PC . A similar examination
was made also by Bottcher and Silbermann 3, Sect. 4.95{4.102], using a dierent
method.
We recall these results for the special case = . For this purpose we introduce
the set T+ = f t 2 T j Im t > 0 g.
Proposition 3.1. Let 2 PC . Then M () is Fredholm if and only if
b () := ( + 0)( + 0) + ( ; 0)( ; 0)(1 ; )
is nonzero for all 2 0 1] and all 2 T+ and
p
b () := ( + 0) + ( ; 0)(1 ; ) ; i ( + 0) ; ( ; 0) (1 ; )
i
On a Class of Toeplitz
+ Hankel Operators
7
is nonzero for all 2 0 1] and all 2 f;1 1g.
Proof. In the above references, the condition for 2 f;1 1g is stated explicitly.
In addition, for 2 T+ one has the requirement that the matrix B () given by
p
0
1
( + 0) + ( ; 0)(1 ; )
( + 0) ; ( ; 0) (1 ; )
@
A
p
( ; 0) ; ( + 0) (1 ; ) ( + 0) + ( ; 0)(1 ; )
be invertible for each 2 0 1]. But it is easy to see that det B () = b ().
Corollary 3.2. Let 2 PC . Then M () is Fredholm if and only if ( 0) 6= 0
for each 2 T and the following is satised:
1 arg ( ; 0)( ; 0) 2= Z + 1 for each 2 T (18)
+
2
( + 0)( + 0)
2
1 arg ( ; 0) 2= Z + for each 2 f;1 1g:
(19)
2
( + 0)
4
Proof. If 2 T+ is xed and runs from 0 to 1, then the value of b () runs
along the line segment from ( ; 0)( ; 0) to ( + 0)( + 0). Similarly, if
2 f;1 1g is xed and runs from 0 to 1, then the value of b () runs along
the half circle with endpoints ( ; 0) and ( + 0) in positive ( = 1) or negative
( = ;1) direction, respectively.
The preceding corollary settles completely the problem of Fredholmness for M ()
with piecewise continuous . However, we have to carry the considerations a little
bit further in order to obtain information about the index of M ().
Theorem 3.3. Let 2 PC . Then M () is Fredholm if and only if can be written
in the form (t) = t{ exp ((t)) with { 2 Z and 2 PC such that the following is
satised :
(i) jIm () + Im ()j < 1=2 for each 2 T+
(ii) ;3=4 < Im 1 () ;< 1=4 and ;1=4 < Im ;1 () < 3=4.
Here () := (2);1 ( ; 0) ; ( + 0) . Moreover, in this case we have
ind M () = ;{, and M () is invertible if { = 0.
Proof. The \if" part is trivial. Now suppose that M () is Fredholm and hence that
the conditions of Corollary 3.2 are satised. For each point 2 T+ f;1 1g one
can choose a piecewise continuous logarithm of which is dened and satises the
conditions (i) and (ii) on an open neighborhood of f g. Because of compactness it
suces to consider only a nite covering of these neighborhoods. By gluing together
the dierent pieces of logarithms of in a suitable way (by altering them locally
by constants of 2iZ if necessary), it is possible to construct a function 2 PC
with = exp that satises the above conditions with the possible exception of
one point, say = ;1. At this point, the size of the jump can be transformed into
the proper value by replacing (t) by (t) ; { log t with suitable { 2 Z. But this
yields the desired representation.
In order to prove the index formula, we rely on the representation
; just considered
and introduce for each 2 0 1] the functions (t) = t{ exp (t) . It easy to
see that also the operators M ( ) are Fredholm. Because the mapping 2 0 1] 7!
M ( ) 2 L(`2 ) is continuous, the index of M ( ) remains constant for 2 0 1].
8
Estelle L. Basor and Torsten Ehrhardt
Note that 1 = and 0 = t{ . So we have ind M () = ind M (t{ ) = ind T (t{ ) =
;{.
In applications, it is often customary to represent a piecewise continuous function
as a certain product. Let t 2 PC with 2 C be the function
(20)
t (e ) = exp(i ( ; ))
0 < < 2:
This function is continuous and nonvanishing on T n f1g and may have a jump at
1 whose size is characterized by t (1 ; 0)=t (1 + 0) = exp(2i ). Each invertible
function 2 PC with nitely many jump discontinuities can be written in the form
i
(e ) = b(e )
(21)
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=1
t r (e ( ; r ) )
i where b is a continuous nonvanishing function and 1 : : : 2 (; ].
Theorem 3.4. Suppose that 2 PC has nitely many jumps. Then M () is
invertible if and only if can be written in the form
R
(22)
Y
(e ) = b(e )t + (e )t ; (e ( ; ) ) t r+ (e ( ; r ) )t r; (e ( + r ) )
R
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=1
i where 1 : : : 2 (0 ) are pairwise distinct points, b is a continuous nonvanishing
function with winding number zero, and
(i) jRe ( + + ; )j < 1=2 for each 1 r R
(ii) ;3=4 < Re + < 1=4 and ;1=4 < Re ; < 3=4.
Proof. As stated in Theorem 3.3, M () is invertible if and only if = exp with the above conditions on . These conditions imply that also has nitely
many jumps, and hence one can always decompose = 1 + 2 where 1 (e )
is continuous and 2 (e ) is piecewise linear with respect to . The factor exp 1
represents the function b, and the factor exp 2 can be identied with the product
of the functions t . Observe that t (e ) is the exponential of a piecewise linear
function. The conditions on the 's are just a restatement of the conditions on the
's.
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Now we proceed with describing the essential spectrum of M (), i.e., the set of
all z 2 C for which M () ; zI is not Fredholm. We introduce the following sets.
Let
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C1(a b) = z 2 C z = a + (b ; a) i (1 ; ) 2 0 1]
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L(a b) = z 2 C z = a + (b ; a) 2 0 1]
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be the half circles and the line segment, respectively, with endpoints a and b.
Obviously, C1 (a a) = L(a a) = fag as a special case. We also dene
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H(a1 a2 b1 b2 ) = z 2 C (a1 ;z )(a2;z )(1;)+(b1;z )(b2;z ) = 0 2 0 1] :
Note that H(a1 a2 b1 b2 ) is symmetric in a1 and a2 as well as in b1 and b2 . In
case of coinciding parameters we simply have H(a1 c b1 c) = L(a1 b1 ) fcg. In
On a Class of Toeplitz
+ Hankel Operators
9
general H is more complicated. If one parameterizes z = x + iy (x y 2 R), then
H(a1 a2 b1 b2 ) is a certain subset of the following cubic curve in x and y:
Im A(x + iy)B (x + iy)
(23)
= 0:
Here A(z ) = (a1 ; z )(a2 ; z ) and B (z ) = (b1 ; z )(b2 ; z ). The entire cubic curve
is obtained by taking in R f1g instead of 0 1]. The reader can easily verify
that the \geometric form" of H may be quite dierent.
Corollary 3.5. Let 2 PC . Then the essential spectrum of M () is equal to
H ( ; 0) ( ; 0) ( + 0) ( + 0) C ( ; 0) ( + 0) :
2T+
2f;1 1g
Proof. This follows from Proposition 3.1. The essential spectrum consists exactly
of the points z 2 C for which there exist a 2 T+ f;1 1g and 2 0 1] such
that b ( ; z ) = 0.
We remark in this connection that the essential range of 2 PC , i.e., the
spectrum of as an element of L1 (T), is just f ( 0) j 2 T g. Hence the
essential range of is always contained in the essential spectrum of M (). In
general this inclusion is proper.
In what follows we give a geometric description of the spectrum of M (). We
will assume for simplicity that has only nitely many jumps and is piecewise
smooth. The general case could be treated with the same ideas, but would require
more detailed explanations.
First of all, the values of (t) as t runs along T (in positive orientation) describe a
certain curve. This curve may be closed if is continuous, or may consist of several
components if has jumps (see Figure 1). Moreover, this curve has a natural
orientation induced from t 2 T.
Now we ll in certain additional pieces as follows. If the function is discontinuous at 2 f;1 1g, then we add the half circle C (( ; 0) ( + 0)). If is
discontinuous at 2 T n f;1 1g, but is continuous at , then we ll in the line
segment L(( ; 0) ( +0)). Finally, if has jumps at both 2 Tnf;1 1g, then
we add H(( ; 0) ( ; 0) ( + 0) ( + 0)). These new pieces constitute themselves oriented curves, where the orientation is now induced from the parameter
2 0 1] that occurs in the denition of these sets.
Gluing together these pieces with the former curve we obtain an oriented curve
# . By Corollary 3.5, the image of # is exactly sp ess M (). Remark again that
the \geometric form" of this curve may be quite dierent. For instance, it may
consist of several closed oriented curves (see Figure 2).
Because # possesses an orientation, it is possible to associate to each point
z 2= im # a winding number wind (# z ). If # consists of several components,
this is just the sum of the usual winding numbers with respect to these components.
These considerations now allow us to describe the spectrum of M ():
(24)
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The proof can be carried out by continuously deforming . We leave the details to
the reader.
10
Estelle L. Basor and Torsten Ehrhardt
1+
-1-
0.5
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i+
-0.5
-i +
-i-
-0.5
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1-
Figure 1.
1
0.5
Image of a piecewise continuous function with four jumps.
0.5
-0.5
0.5
1
-0.5
Essential spectrum of M () with corresponding to
the function in Figure 1.
Figure 2.
The gures above give an example of the image of a piecewise continuous curve
and its (essential) spectrum. The example comes from a product of four of the
standard t functions, with jumps at 1 ;1 i and ;i and values of equal to
1=3 ;3=4 5=12 and 1=3 respectively and a normalizing factor of ;e; 24 for picture purposes only. In this example, the spectrum consists of the area bounded by
the curve in Figure 2. As one can see the spectrum as well as the essential spectrum
of this operator is not connected, contrasted to the Toeplitz case.
i=
On a Class of Toeplitz
+ Hankel Operators
11
Stability of the nite section method for P C symbols
4.
If one considers the nite truncations of the innite matrix M (), one obtains
the sequence fM ()g1=1 of n n matrices, the so-called nite sections. This
sequence is said to be stable if the matrices are invertible for suciently large n
and if their inverses (considered as operators in L(`2 )) are uniformly bounded.
The problem of stability occurs naturally when examining, for instance, the
question whether the inverses of M () converge strongly on `2 to the inverse of
M (). This question is obviously of interest in numerical analysis, but it also
emerges in the investigation of the asymptotic behavior of the determinants of
M () (see 2]).
Employing Banach algebra methods, necessary and sucient conditions for the
stability of sequences of the form fT () + H ()g1=1 with 2 PC were established by Roch and Silbermann in 7, Theorem 5.3]. These conditions say that the
sequence is stable if and only if the operators T () + H () and T (e) are invertible
and if certain operators A (associated to each point 2 T+ f;1 1g) are invertible. Spitkovsky and Tashbaev pointed out that the invertibility of the operators
A can be reduced to the factorization of certain 2 2 matrix functions, and above
all they solved the corresponding factorization problem 8].
Invertibility criteria for Toeplitz operators with piecewise continuous symbols are
well known 3]. But the invertibility of Toeplitz + Hankel operators in the general
case remains still unsolved.
We proceed with recalling the stability results 7, 8] for the special case = .
Proposition 4.1. Let 2 PC . Then the sequence fM ()g1=1 is stable if and
only if the operators M () and T (e) are invertible and the following conditions are
satised :
(i) j ()j < 1=2, j ()j < 1=2 and j () + ()j < 1=2 for each 2 T+
(ii) ;1=2 < 1 () < 1=4 and ;1=4 < ;1 () < 1=2.
Here () := (2);1 arg (( ; 0)=( + 0)) where the argument is chosen in
(; ].
Proof. As said before, one has to analyze the invertibility of certain operators A
with 2 T+ f;1 1g. We refer to 8, Theorem 4] and to the proof given there. It
is stated that the operators A are invertible if and only if ( 0) 6= 0 for each
2 T (which is already a consequence of the invertibility of M () or of T (e)) and
if the following conditions A) and B) are fullled:
A) For 2 T+, the values ( ) = ( ; 0)=( +0) and ( ) = ( ; 0)=( +0)
are not negative, and if 1 and 2 are the zeros of the quadratic polynomial
0)( ; 0) + 1 + ( ; 0)( ; 0) 2 ; (( ;
+ 0)( + 0)
( + 0)( + 0)
then 1 and 2 are not negative, and
0 = arg ( ) + arg ( ) ; arg 1 ; arg 2 where the arguments are chosen in (; ). However, the zeros of the quadratic
polynomial are just 1 = 1 and 2 = ( )( ). Now it is easily seen that this is
equivalent to (i).
n
n
n
n
n
n
n
n
n
12
Estelle L. Basor and Torsten Ehrhardt
B) For 2 f;1 1g, zero does not lie in the closed half disk that has the boundary
L ( ; 0) ( + 0) C ( ; 0) ( + 0) :
Again this is equivalent to (ii).
Note that the preceding result completely solves the stability problem for sequences fM ()g1=1 with 2 PC . We now want to point out what stability
means for functions represented in the form (21).
Theorem 4.2. Suppose that 2 PC has nitely many jumps. Then fM ()g1=1
is stable if and only if can be written in the form
n
n
n
(25)
n
Y
(e ) = b(e )t + (e )t ; (e ( ; ) ) t r+ (e ( ; r ) )t r; (e ( + r ) )
R
i
i
i
i i r
=1
i where 1 : : : 2 (0 ) are pairwise distinct points, b is a continuous nonvanishing
function with winding number zero, and
(i) jRe + j < 1=2, jRe ; j < 1=2 and jRe ( + + ; )j < 1=2 for each 1 r R
(ii) ;1=2 < Re + < 1=4 and ;1=4 < Re ; < 1=2.
R
r
r
r
r
Proof. First of all note that T (e) is invertible if and only if so is T (). This, in
turn, is equivalent to the condition that can be written in the form (25) such
that the moduli of the real parts of the 's are less than 1=2 and that the winding
number of b is zero 3].
Now assume that M () is stable. It is not too hard to see that the conditions
(i) and (ii) of Proposition 4.1 imply that can be represented in the form (25)
by choosing values of the 's corresponding to the appropriate (): The winding
number condition on b follows from the invertibility of M () or T (). Note that
the reverse implication is trivial.
n
Comparing the preceding theorem with Theorem 3.4, we can draw the immediate
conclusion that | in contrast to the (scalar) Toeplitz case | the invertibility of
M () does in general not guarantee the stability of the sequence fM ()g1=1 of
nite sections.
Another point is that we had to use in the proof only the invertibility of either
M () or T (). This has its explanation in the following. \If 2 PC is invertible
and satises conditions (i) and (ii) of Proposition 4.1, then both M () and T ()
are Fredholm and ind M () = ind T (). In particular, under these assumptions,
M () is invertible if and only if so is T ()." We omit a proof of these assertions,
which can be carried out by similar arguments as in the proof of Theorem 3.3.
In contrast, it is possible that both M () and T () are invertible but conditions
(i) and (ii) of Proposition 4.1 are not fullled (hence fM ()g1=1 is not stable).
We give two concrete examples of functions and note that representation (21) is
not unique. The rst example is
n
n
(e ) = t + (e )t ; (e ( ; ))
= ;t +;1 (e )t ; +1 (e ( ; ) )
i
(26)
i
i
i i n
n
On a Class of Toeplitz
+ Hankel Operators
13
where Re + 2 (1=4 1=2) and Re ; 2 (;1=2 ;1=4). The second example is
(e ) = t 1+ (e ( ; 1 ) )t 1; (e ( + 1 ) )t 2+ (e ( ; 2 ) )t 2; (e ( + 2 ) )
(27)
= e ( 2 ; 1 ) t 1+;1 (e ( ; 1) )t 1; (e ( + 1) )t 2+ +1 (e ( ; 2 ) )t 2; (e ( + 2 ) )
where 1 2 2 (0 ), 1 6= 2 , Re 1+ 2 (0 1=2), Re 1; 2 (0 1=2), Re (1+ + 1; ) 2
(1=2 1), Re 2+ 2 (;1=2 0), Re 2; 2 (;1=2 0) and Re (2+ + 2; ) 2 (;1 ;1=2).
In both examples, the rst and second representation guarantee the invertibility of
T () and M (), respectively.
i
i i i i i i i i i Stability for approximate identities of P C symbols
5.
Let 0 1) be an unbounded index set. Given a (generalized) sequence
f g 2 of functions in L1 (T), one can consider the sequence fM ( )g 2 of
operators. This sequence is said to be stable if
(i) There exists a 0 2 0 1) such that the operators M ( ) are invertible for
each 2 0 1) \ (ii) sup 2 0 1)\ kM ( );1 kL( 2) is nite.
It is, of course, hopeless to examine the stability problem for arbitrary such sequences. We will therefore restrict ourselves to certain classes that will be described
below.
Let K be a function in L1 (R) which satises
`
Z1
(28)
;1
K (x) dx = 1:
For 2 0 1) we introduce the linear bounded mappings k : L1 (T) ! L1(T)
dened by
(k )(e ) =
(29)
ix
Z1
;1
(e ( ; ) )K (y) dy:
i x
y
The mapping k is called the approximate identity generated by K . A typical
example of an approximate identity is the harmonic extension h dened by
(30)
1
X
h :
n
=;1
1
X
j je e 7!
inx
=;1
n
n
inx
n
0 < 1:
n
The use of the index instead of should not cause
; confusion.
The relationship
is constituted through h = k with K (x) = 1= (1 + x2 ) and = ;1= log .
Other examples of approximate identities are the Fejer-Cesaro means and the moving average 1]. We remark that stability of Toeplitz operators fT (k )g for quite
general types of approximating identities was rst examined by Bottcher and Silbermann 4] (see also 3, Chap. 3]).
For a given approximate identity k generated by K we dene a function f by
ix Z
(31)
f 11 +
=
K (y) dy x 2 R:
; ix
;1
Note that f 2 PC;1 , where PC;1 denotes the set of functions in PC which are
continuous on T n f;1g. In particular f (;1 + 0) = 0 and f (;1 ; 0) = 1. Remark
that in most cases of interest (e.g., if K (x) 0), the spectrum of f in L1 (T) is
just 0 1].
x
14
Estelle L. Basor and Torsten Ehrhardt
For an invertible function 2 PC;1 , the Cauchy index is dened as follows:
; (32)
ind = 21 arg e =;
where arg is chosen in PC;1 . Given a function 2 PC and 2 T we introduce
for a xed approximate identity with corresponding function f , the function
(33)
(e ) = ( + 0)f (e ) + ( ; 0)(1 ; f (e )):
Note that 2 PC;1 and (;1 0) = ( 0).
It is the goal of what follows to investigate sequences fM (k )g 2 with 2
PC . In 5], various sequences of convolution type operators with such generating
functions have been examined in view of stability, including sequences of the form
fT (k ) + H (k )g 2 . The stability criterion established there says that the
latter sequence is stable if and only if certain associated operators are invertible.
Unfortunately, these invertibility problems could not be handled in general. We
will indicate next how the special case = can be treated.
Theorem 5.1. Let k be an approximate identity and let 2 PC . Then the
sequence fM (k )g 2 is stable if and only if M () is invertible, for each 2 T
the functions 2 PC;1 are invertible and the following is satised :
(i) jind + ind j < 1=2 for each 2 T+
(ii) ;1=4 < ind 1 < 3=4 and ;3=4 < ind ;1 < 1=4.
Proof. It can be derived from 5, Corollary 3.2] that fM (k )g 2 is stable if and
only if the operators M (), A = T ( ) + H ( ) for 2 f;1 1g and
L( )P + Q L( )Q A =
L(f )P L(f )Q + P
for 2 T+ are invertible. As to the operator A , it is easily seen that
:
;P ;
P
Q
A = I0 I0 + LL((f )) ;
Q
Hence the invertibility of A is equivalent to the invertibility of the singular integral operator
L( ) ; P ;
f
I + P Q
L(f ) ; Q = PL( ) + QL( ):
This, in turn, is equivalent to saying that the functions and are invertible in
;1
;1
PC;1 and that jind ( f )j < 1=2. Now observe that ind ( f ) = ind ;
ind f = ind + ind .
If = 1, then A1 = M (1 ). The invertibility can be analyzed by Theorem 3.3.
It follows that the function with 1 = exp appearing there is also in PC;1 .
Hence Im = arg 1 up to a constant and by denition Im ;1 () = ind 1 .
If = ;1, then A;1 = WM (b;1 )W where b;1 (t) ;= ;1 (;t),; t 2 T, is the
\rotated" function and W 2 L(`2 ) is the operator W : x 0 7! (;1) x 0 .
So we are left with the invertibility of M (b;1 ). We can argue as above, and it follows
similarly that Im = arg b;1 up to a constant. Hence Im 1 () = ind ;1 .
i
i
i
i
k
k
k
k
k
On a Class of Toeplitz
+ Hankel Operators
15
The previous result answers the question about stability for the case under consideration completely, and in general no further essential simplications are possible.
In the special case where the spectrum of f is just 0 1], the situation is dierent.
Corollary 5.2. Let k be an approximate identity with sp 1(T)f = 0 1] and let
2 PC . Then the sequence fM (k )g 2 is stable if and only if M () is invertible
and the conditions (i) and (ii) of Proposition 4.1 are satised.
Proof. We remark that is invertible if and only if ( 0) 6= 0 and j ()j < 1=2.
Moreover, () = ;ind by denition.
The next result discusses the stability condition in terms of the product representation. The proof can be carried out in the same way as the proof of Theorem 4.2.
Corollary 5.3. Let k be an approximate identity with sp 1(T)f = 0 1]. Assume
that 2 PC has nitely many jumps. Then fM (k )g 2 is stable if and only if
can be written in the form and with the conditions stated in Theorem 4.2.
We can conclude that the invertibility of M () does in general not guarantee the
stability of fM (k )g 2 . Again this contrasts the Toeplitz case 5, Sect. 3.3].
Now we turn our attention to another type of sequences. They are obtained
by considering exponentials of approximate identities of piecewise continuous functions.
Theorem 5.4. Let k be an approximate identity and let 2 PC . Then the
sequence fM (exp(k ))g 2 is stable if and only if fullls the conditions (i) and
(ii) stated in Theorem 3.3.
Proof. This can be veried by combining Theorem 2.2 and Proposition 3.4 of 5]
(see also 5, Corollary 3.5] for a related situation). One arrives at similar invertibility
conditions as stated in Theorem 5.1 above. The only dierence is that one has to
replace by exp and by exp . The argumentation is analogous. Finally
note that exp 2 PC;1 and ind exp = Im ;1 ( ) = ;Im (). Moreover,
the conditions (i) and (ii) of Theorem 3.3 ensure the invertibility of M (exp ).
This theorem (in conjunction with Theorem 3.3) has an interesting consequence.
Suppose that M () with 2 PC is invertible. Then one can always nd a 2 PC
with = exp such that the sequence fM (exp(k ))g 2 is stable. One just has
to choose with the conditions stated in Theorem 3.3. Observe that if (the
logarithm of ) is not chosen \properly", then the sequence fails to be stable. The
constructed sequence is a so-called approximating sequence for the operator M ()
in the sense that M (exp(k )) ! M () strongly on `2 as ! 1.
A special case of the previous theorem, which is of interest in applications 2],
is considered next. Let t with 0 < 1 and 2 C be the (smooth) function
dened by
;
;
(34)
t (e ) = 1 ; e; ; 1 ; e :
Corollary 5.5. The sequence fM ( )g0 1 dened by
L
L
i
(35)
(e ) = b(e )t
i
i
+ i
(e )t
i
<
; (ei(;) )
Y
R
r
=1
i
t
r+ (
e ( ; r ) )t
i e ( + r))
r; (
i 16
Estelle L. Basor and Torsten Ehrhardt
where 1 : : : 2 (0 ) are pairwise distinct points and b is a continuous nonvanishing function with winding number zero, is stable (as ! 1 ; 0) if and only if
the parameters satisfy the conditions (i) and (ii) of Theorem 3.4.
Proof. The functions t are the exponentials of the harmonic extensions h of
the piecewise linear functions log t (e ) = i ( ; ), 0 < < 2. Hence the
assertion can be deduced for functions (35) with b replaced by b := exp (h log b).
Because b ! b in the norm of L1 (T), the actual and the modied sequence of
operators dier only by a sequence tending to zero in the operator norm.
R
i
References
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Zbl 605.47025.
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Department of Mathematics, California Polytechnic State University, San Luis
Obispo, CA 93407
ebasor@calpoly.edu
Fakultat fur Mathematik, Technische Universitat Chemnitz, 09107 Chemnitz, Germany
tehrhard@mathematik.tu-chemnitz.de
This paper is available via http://nyjm.albany.edu:8000/j/1999/5-1.html.