UPPER AND LOWER BOUNDS FOR REGULARIZED
DETERMINANTS
Upper and Lower Bounds
For Regularized Determinants
M. I. GIL’
Department of Mathematics
Ben Gurion University of the Negev
P.0. Box 653, Beer-Sheva 84105, Israel
EMail: gilmi@cs.bgu.ac.il
M. I. Gil’
vol. 9, iss. 1, art. 2, 2008
Title Page
Contents
Received:
11 January, 2007
Accepted:
21 January, 2008
Communicated by:
F. Hansen
2000 AMS Sub. Class.:
47B10.
Key words:
von Neumann-Schatten ideal, Regularized determinant.
Abstract:
Let Sp be the von Neumann-Schatten ideal of compact operators in a separable
Hilbert space. In the paper, upper and lower bounds for the regularized determinants of operators from Sp are established.
Acknowledgements:
This research was supported by the Kamea fund of the Israel.
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Contents
1
Upper bounds
3
2
Lower Bounds
9
Upper and Lower Bounds
For Regularized Determinants
M. I. Gil’
vol. 9, iss. 1, art. 2, 2008
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Contents
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1.
Upper bounds
For an integer p ≥ 2, let Sp be the von Neumann-Schatten ideal of compact operators
A in a separable Hilbert space with the finite norm Np (A) = [Trace(AA∗ )p/2 ]1/p
where A∗ is the adjoint. Recall that for an A ∈ Sp the regularized determinant is
defined as
#
" p−1
∞
Y
X λm
(A)
j
detp (A) :=
(1 − λj (A)) exp
m
m=1
j=1
where λj (A) are the eigenvalues of A with their multiplicities arranged in decreasing
order.
The inequality
(1.1)
detp (A) ≤
Upper and Lower Bounds
For Regularized Determinants
M. I. Gil’
vol. 9, iss. 1, art. 2, 2008
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exp[qp Npp (A)]
Contents
N22 (A)/2
is well-known, cf. [2, p. 1106], [4, p. 194]. Recall that | det2 (A)| ≤ e
, cf.
[5, Section IV.2 ]. However, to the best of our knowledge, the constant qp for p > 2
is unknown in the available literature although it is very important, in particular, for
perturbations of determinants. In the present paper we suggest bounds for qp (p >
2). In addition, we establish lower bounds for detp (A). As far as we know, the lower
bounds have not yet been investigated in the available literature.
Our results supplement the very interesting recent investigations of the von NeumannSchatten operators [1, 3, 8, 9, 10]. In connection with the recent results on determinants, the paper [6] should be mentioned. It is devoted to higher order asymptotics
of Toeplitz determinants with symbols in weighted Wienar algebras.
To formulate the main result we need the algebraic equation
"
#
p−3
m
X
x
(p > 2).
(1.2)
xp−2 = p(1 − x) 1 +
m+2
m=1
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Below we prove that it has a unique positive root x0 < 1. Moreover,
r
p
(1.3)
x0 ≤ p−2
.
p+1
Theorem 1.1. Let A ∈ Sp (p = 3, 4, . . . ). Then inequality (1.1) holds with
qp =
1
.
p(1 − x0 )
Upper and Lower Bounds
For Regularized Determinants
The proof of this theorem is divided into a series of lemmas presented below.
Lemma 1.2. Equation (1.2) has a unique positive root x0 < 1.
M. I. Gil’
vol. 9, iss. 1, art. 2, 2008
Proof. Rewrite (1.2) as
p−2
g(x) :=
x
−
p(1 − x)
p−1 m−2
X
x
1+
m
m=3
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!
= 0.
Clearly, g(0) = −1, g(x) → +∞ as x → 1 − 0. So (1.2) has at least one root from
(0, 1). But from (1.2) it follows that a root from [1, ∞) is impossible. Moreover,
(1.2) is equivalent to the equation
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p−1
X xm−p
1
1
= p−2 +
.
p(1 − x)
x
m
m=3
The left part of this equation increases and the right part decreases on (0, 1). So the
positive root is unique.
Furthermore, consider the function
"
p−1 m
X
z
f (z) := Re ln(1 − z) +
m
m=1
#
(z ∈ C; p > 2).
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Clearly,
f (z) = − Re
∞
X
zm
m
m=p
(|z| < 1).
Lemma 1.3. Let w ∈ (0, 1). Then
|f (z)| ≤
rp
p(1 − w)
Upper and Lower Bounds
For Regularized Determinants
(r ≡ |z| < w).
M. I. Gil’
Proof. Clearly,
vol. 9, iss. 1, art. 2, 2008
∞
X
rm
|f (z)| ≤
m
m=p
(r < 1).
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Consequently,
Contents
Z
|f (z)| ≤
∞
r X
sm−1 ds =
0 m=p
Z
0
r
sp−1
∞
X
sk ds =
Z
0
k=0
r
sp−1 ds
.
1−s
Hence we get the required result.
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Lemma 1.4. For any w ∈ (0, 1) and all z ∈ C with |z| ≥ w, the following inequality
is valid:
"
#
p−1
m
X
w
|f (z)| ≤ hp (w)rp where hp (w) = w−p w2 +
(p > 2).
m
m=3
Proof. Take into account that
2
|(1 − z)ez |2 = (1 − 2 Re z + r2 )e2x ≤ e−2 Re z+r e2 Re z = er
2
(z ∈ C),
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since 1 + x ≤ ex , x ∈ R. So
"
" p−1
#
#
p−1 m
m X
X
r
z
.
(1 − z) exp
≤ exp r2 +
m
m
m=1
m=3
Therefore,
p−1 m
X
r
|f (z)| ≤ r +
m
m=3
2
But
Upper and Lower Bounds
For Regularized Determinants
(z ∈ C).
M. I. Gil’
vol. 9, iss. 1, art. 2, 2008
"
#
p−1 m
X
r
r−p ≤ hp (w)
r2 +
m
m=3
(r ≥ w).
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This proves the lemma.
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Lemmas 1.3 and 1.4 imply
Corollary 1.5. One has
|f (z)| ≤ q̃p rp (z ∈ C, p > 2)
where
q̃p := min max hp (w),
w∈(0,1)
However, function hp (w) decreases in w ∈ (0, 1) and
minimum in the previous corollary is attained when
1
p(1−w)
1
p(1 − w)
This equation is equivalent to (1.2). So q̃p = qp and we thus get the inequality
|f (z)| ≤ qp rp
(z ∈ C).
.
increases. So the
1
hp (w) =
.
p(1 − w)
(1.4)
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Lemma 1.6. Let A ∈ Sp , p > 2. Then detp (A) ≤ exp[qp wp (A)] where
wp (A) :=
∞
X
|λk (A)|p .
k=1
Proof. Due to (1.4),
detp (A) ≤
∞
Y
"
eqp |λj
(A)|p
≤ exp
j=1
∞
X
#
qp |λj (A)|p .
k=1
Upper and Lower Bounds
For Regularized Determinants
M. I. Gil’
vol. 9, iss. 1, art. 2, 2008
As claimed.
Proof of Theorem 1.1. The assertion of Theorem 1.1 follows from the previous lemma
and the inequality
∞
X
|λj (A)|p ≤ Npp (A)
k=1
cf. [5].
≤ p(1 − x0 )
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Furthermore, from (1.2) it follows that
xp−2
0
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p−3
X
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xm
0 = p(1 −
xp−2
0 )
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m=0
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since
p−3
X
m=0
xm
0 =
1 − xp−2
0
.
1 − x0
This proves inequality (1.3). Thus
qp ≤
1
p 1−
q
p−2
p
p+1
.
Note that if the spectral radius rs (A) of A is less than one, then according to Lemma
1.3 one can take
1
qp =
.
p(1 − rs (A))
Corollary 1.7. Let A, B ∈ Sp (p > 2). Then
| detp (A) − detp (B)| ≤ Np (A − B) exp[qp (1 + Np (A) + Np (B))p ].
Indeed, this result is due to Theorem 1.1 and the theorem by Seiler and Simon [7]
(see also [4, p. 32]).
Upper and Lower Bounds
For Regularized Determinants
M. I. Gil’
vol. 9, iss. 1, art. 2, 2008
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2.
Lower Bounds
In this section for brevity we put λj (A) = λj . Denote by L a Jordan contour
connecting 0 and 1, lying in the disc {z ∈ C : |z| ≤ 1}, not containing the points
1/λj for any eigenvalue λj , such that
φA :=
(2.1)
inf
s∈L; k=1,2,...
|1 − sλk | > 0.
Upper and Lower Bounds
For Regularized Determinants
Let l = |L| be the length of L. For example, if A does not have eigenvalues on
[1, ∞), then one can take L = [0, 1]. In this case l = 1 and φA = inf k,s∈[0,1] |1−sλk |.
If rs (A) < 1, then l = 1, φA ≥ 1 − rs (A).
vol. 9, iss. 1, art. 2, 2008
Theorem 2.1. Let A ∈ Sp (p = 2, 3, . . . ), 1 6∈ σ(A) and condition (2.1) hold. Then
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−
| detp (A)| ≥ e
p
lNp (A)
φA
Contents
.
Proof. Consider the function
D(z) =
∞
Y
Gj (z)
where
j=1
Clearly,
" p−1
#
X z m λm
j
Gj (z) := (1 − zλj ) exp
.
m
m=1
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D0 (z) =
∞
X
G0k (z)
"
G0j (z) = −λj + (1 − zλj )
∞
Y
Gj (z)
j=1,j6=k
k=1
and
M. I. Gil’
p−2
X
m=0
#
z m λm+1
j
#
p
X
z m λm
j
exp
.
m
m=1
"
Close
But
−λj + (1 − zλj )
p−2
X
z m λm+1
= −z p−1 λpj ,
j
m=0
since
p−2
X
z m zjm =
m=0
So
1 − (zλj )p−1
.
1 − zλj
p
X
z m λm
j
G0j (z) = −z p−1 λpj exp
m
m=1
"
#
Upper and Lower Bounds
For Regularized Determinants
z p−1 λpj
=−
Gj (z).
1 − zλj
Hence, D0 (z) = h(z)D(z), where
h(z) := −z p−1
λpk
1 − zλk
Contents
.
Consequently,
Z
D(1) = detp (A) = exp
h(s)ds .
L
But |s| ≤ 1 for any s ∈ L and thus
Z
X
Z
∞
|s|p−1 |ds|
p
h(s)ds ≤
λ
≤ wp (A)lφ−1
A .
k
|1
−
sλ
|
k
L
L
k=1
Therefore,
Z
Z
|detp (A)| = exp
h(s)ds] ≥ exp − h(s)ds ≥ exp[−wp (A)lφ−1
A ].
This proves the theorem.
vol. 9, iss. 1, art. 2, 2008
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∞
X
k=1
L
M. I. Gil’
L
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References
[1] P. CHAISURIYA AND SING-CHEONG ONG, Schatten’s theorems on functionally defined Schur algebras, Int. J. Math. Math. Sci., (14) 2005, 2175–2193.
[2] N. DUNFORD AND J.T. SCHWARTZ, Linear Operators, Part II. Spectral Theory. Interscience Publishers, New York, London, 1963.
[3] L.G. GHEORGHE, Hankel operators in Schatten ideals, Ann. Mat. Pura Appl.,
IV Ser., 180(2) (2001), 203–210.
[4] I. GOHBERG, S. GOLBERG AND N. KRUPNIK, Traces and Determinants of
Linear Operators, Birkháuser Verlag, Basel, 2000.
[5] I.C. GOHBERG AND M.G. KREIN, Introduction to the Theory of Linear Nonselfadjoint Operators, Trans. Mathem. Monographs, v. 18, Amer. Math. Soc.,
Providence, R. I., 1969.
Upper and Lower Bounds
For Regularized Determinants
M. I. Gil’
vol. 9, iss. 1, art. 2, 2008
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Contents
[6] A. Yu KARLOVICH, Higher order asymptotics of Toeplitz determinants with
symbols in weighted Wienar algebras, Math. Analysis Appl., 320 (2006), 944–
963.
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[7] E. SEILER AND B. SIMON, An inequality for determinants, Proc. Nat. Acad.
Sci., 72 (1975), 3277–3278.
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[8] M. SIGG, A Minkowski-type inequality for the Schatten norm, J. Inequal. Pure
Appl. Math., 6(3) (2005), Art. 87. [ONLINE: http://jipam.vu.edu.
au/article.php?sid=560].
[9] M.W. WONG, Schatten-von Neumann norms of localization operators, Arch.
Inequal. Appl., 2(4) (2004), 391–396.
[10] JINGBO XIA, On the Schatten class membership of Hankel operators on the
unit ball, Ill. J. Math., 46(3) (2002), 913–928.
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