Estimating the number of eigenvalues of linear operators on Banach spaces

Demuth, Michael; Hanauska, Franz; Hansmann, Marcel; Katriel, Guy

doi:10.1016/j.jfa.2014.11.007

Cited by 7 publications

(27 citation statements)

References 29 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The previous estimate broadly generalizes a corresponding result in , which considered the case where

I = {scriptS}_{p}^{(a)} (X)

.…”

Section: Eigenvalues Of Compactly Perturbed Operatorssupporting

confidence: 80%

“…So we are in a position to apply Theorem : a conformal mapping ϕ that maps Ω onto

double-struckD

(and ∞ onto 0) is given by

ϕ (λ) = t / λ

, so

r_{Ω} (Ω^{'}) = \underset{λ \in Ω^{'}}{trueprefixsup} | ϕ (λ) | = \frac{t}{s} .

We thus obtain from Theorem that for

s > t > R

n_{A + K} (s) \leq \frac{γ_{p}^{p} Γ_{p}}{ε^{p} log (\frac{1}{r_{Ω} (Ω^{'})})} {∥ K ∥}_{I}^{p} = \frac{C_{A}^{p} γ_{p}^{p} Γ_{p}}{{(t - R)}^{p} log (\frac{s}{t})} {∥ K ∥}_{I}^{p} .

All that remains is to maximize the function

f (t) = {(t - R)}^{p} log (\frac{s}{t}), t \in (R, s)

. This had already been done in (see the computations preceeding Theorem 4.2 in that paper), where it was shown that

\underset{t \in (R, s)}{trueprefixmax} f (t) = \frac{s^{p}}{Φ_{p} (\frac{R}{s})} .

This shows the validity of . The validity of follows from and estimate .

…”

Section: Eigenvalues Of Compactly Perturbed Operatorsmentioning

confidence: 99%

“…We are interested in the discrete spectrum of the perturbed operator

A + K

. Remark Our approach in this section follows along the lines of the approach developed in . However, we note that in the authors considered the case where

I = {scriptS}_{p}^{(a)} (X)

is the approximation number ideal only.…”

Section: Eigenvalues Of Compactly Perturbed Operatorsmentioning

confidence: 99%

“…In order to formulate the next theorem, let us introduce a function

Φ_{p} : (0, 1) \to R

given by

Φ_{p} (x) = \frac{{(W ((), \frac{1}{p}, e^{\frac{1}{p}}, x))}^{p}}{{(() \frac{1}{p} - W (\frac{1}{p} e^{\frac{1}{p}} x))}^{p + 1} x^{p}},

where

W : [0, \infty) \to [0, \infty)

is the Lambert W‐function defined by

W (x) e^{W (x)} = x

. This function satisfies the bound

Φ_{p} (x) \leq \frac{{(p + 1)}^{p + 1}}{p^{p}} \cdot \frac{1}{{(1 - x)}^{p + 1}}, 0 < x < 1,

as has been shown in (see the proof of Corollary 4.3 in that paper). Theorem Suppose that for some

R \geq r (A)

and

C_{A} > 0

we have

‖{(λ - A)}^{- 1}‖ \leq \frac{C_{A}}{| λ | - R}, | λ | > R .

Then for

s > R

the following holds:

n_{A + K} (s) \leq C_{A}^{p} γ_{p}^{p} Γ<...>

…”

Section: Eigenvalues Of Compactly Perturbed Operatorsmentioning

confidence: 99%

“…Our main impetus for the present work came from the recent paper , which studied the distribution of eigenvalues of compactly perturbed operators on Banach spaces. One of the main results of reads as follows: Given a bounded operator

A \in B (X)

and a compact operator K on X , let

n_{A + K} (s)

denote the number of discrete eigenvalues of

A + K

{λ \in C : | λ | > s}

. Then the following holds:

n_{A + K} (s) \leq C_{p} \frac{s}{{(s - ∥ A ∥)}^{p + 1}} \sum_{j \in N} a_{j} {(K)}^{p}, s > ∥ A ∥,

where

a_{j} (K)

denotes the j th approximation number of K .…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Perturbation determinants in Banach spaces — with an application to eigenvalue estimates for perturbed operators

Hansmann

2016

Mathematische Nachrichten

View full text Add to dashboard Cite

In the first part of this paper we provide a self‐contained introduction to (regularized) perturbation determinants for operators in Banach spaces. In the second part, we use these determinants to derive new bounds on the discrete eigenvalues of compactly perturbed operators, broadly extending some recent results by Demuth et al. In addition, we also establish new bounds on the discrete eigenvalues of generators of C0‐semigroups.

show abstract

“…The previous estimate broadly generalizes a corresponding result in , which considered the case where

I = {scriptS}_{p}^{(a)} (X)

.…”

Section: Eigenvalues Of Compactly Perturbed Operatorssupporting

confidence: 80%

“…So we are in a position to apply Theorem : a conformal mapping ϕ that maps Ω onto

double-struckD

(and ∞ onto 0) is given by

ϕ (λ) = t / λ

, so

r_{Ω} (Ω^{'}) = \underset{λ \in Ω^{'}}{trueprefixsup} | ϕ (λ) | = \frac{t}{s} .

We thus obtain from Theorem that for

s > t > R

n_{A + K} (s) \leq \frac{γ_{p}^{p} Γ_{p}}{ε^{p} log (\frac{1}{r_{Ω} (Ω^{'})})} {∥ K ∥}_{I}^{p} = \frac{C_{A}^{p} γ_{p}^{p} Γ_{p}}{{(t - R)}^{p} log (\frac{s}{t})} {∥ K ∥}_{I}^{p} .

All that remains is to maximize the function

f (t) = {(t - R)}^{p} log (\frac{s}{t}), t \in (R, s)

. This had already been done in (see the computations preceeding Theorem 4.2 in that paper), where it was shown that

\underset{t \in (R, s)}{trueprefixmax} f (t) = \frac{s^{p}}{Φ_{p} (\frac{R}{s})} .

This shows the validity of . The validity of follows from and estimate .

…”

Section: Eigenvalues Of Compactly Perturbed Operatorsmentioning

confidence: 99%

“…We are interested in the discrete spectrum of the perturbed operator

A + K

. Remark Our approach in this section follows along the lines of the approach developed in . However, we note that in the authors considered the case where