Entropy numbers and eigenvalues of operators Radosaw Szwedek Adam - - PowerPoint PPT Presentation

Entropy numbers and eigenvalues of operators

Radosław Szwedek

Adam Mickiewicz University in Poznań Faculty of Mathematics and Computer Science

Aleksander Pełczyński Memorial Conference 2014 The talk is based on a recent work with Mieczysław Mastyło

Spectral radius formula

◮ The spectrum σ(T) of T on a complex Banach space X

σ(T) :=

• λ ∈ C ; λIX − T is not invertible in L(X)
• ◮ The essential spectrum

σess(T) := σ(T) T is the coset of T in the Calkin algebra L(X)/K(X).

◮ Gelfand’s spectral radius formula

r(T) := sup

λ∈σ(T)

|λ| = lim

m→∞ T m1/m ◮ The essential spectral radius

ress(T) := r(T) = lim

m→∞ T m1/m ess

Eigenvalue sequence

The Riesz part of the spectrum Λ(T) is at most countable and consists of isolated eigenvalues of finite algebraic multiplicity. Λ(T) :=

• λ ∈ σ(T) ; |λ| > ress(T)
• We assign an eigenvalue sequence
• λn(T)

∞

n=1 for T ∈ L(X) from

the elements of the set Λ(T) ∪

• ress(T)
• as follows:

◮ The eigenvalues are arranged in an order of non-increasing

absolute values.

◮ Every eigenvalue λ ∈ Λ(T) is counted according to its

algebraic multiplicity.

◮ If T possesses less than n eigenvalues λ with |λ| > ress(T), we

let λn(T) = λn+1(T) = . . . = ress(T)

Entropy numbers

Definition

The n-th entropy number εn(T) of T ∈ L(X, Y ) is defined by εn(T) := inf

• ε > 0 ; T(UX) ⊂

n

• i=1

{yi + εUY } , yi ∈ Y

• ◮ Entropy numbers are monotonne

0 . . . ε3(T) ε2(T) ε1(T) = T

◮ The measure of non-compactness

β(T) := lim

n→∞ εn(T)

Carl-Triebel’s inequality (1980)

Let

• λn(T)

∞

n=1 be an eigenvalue sequence of T ∈ L(X) on

a complex Banach space X.

◮ Carl’s inequality

|λn(T)| √ 2 en(T) where en(T) := ε2n−1(T)

◮ Carl-Triebel’s inequality

n

• i=1

|λi(T)| 1/n inf

k∈N k1/(2n)εk(T)

Banach couple

◮ We call

A := (A0, A1) a Banach couple if both A0 and A1 are Banach spaces such that A0, A1 ֒ → X For a given Banach couple A, we define spaces

◮ intersection A0 ∩ A1 with the norm

aA0∩A1 = max

• aA0 , aA1
• ◮ sum A0 + A1 with the norm

aA0+A1 = inf

a=a0+a1

• a0A0 + a1A1

Interpolation functor

◮ By T :

A → B we denote an operator T : A0 + A1 → B0 + B1, such that T|Aj ∈ L(Aj, Bj), j = 0, 1

Definition

By an interpolation functor we mean a mapping F : B → B

◮ A0 ∩ A1 ⊂ F(

A) ⊂ A0 + A1 for any A ∈ B

◮ T

• F(

A)

• ⊂ F(

B) for any

• A,

B ∈ B and T : A → B

Interpolation functor of exponential type of θ

For all interpolation functors F TF(

A)→F( B) C max

• TA0→B0 , TA1→B1
• If in addition there exists θ ∈ (0, 1) such that

TF(

A)→F( B) C T1−θ A0→B0 Tθ A1→B1 ,

then F is called of exponential type of θ.

◮ The real F(·) = (·)θ,q and complex F(·) = [·]θ interpolation

functors are of exponential type of θ.

Recollect...

◮ The n-th entropy number εn(T) of T ∈ L(X, Y )

εn(T) := inf

• ε > 0 ; T(UX) ⊂

n

• i=1

{yi + εUY } , yi ∈ Y

• ◮ The measure of non-compactness

β(T) := lim

n→∞ εn(T)

Interpolation of the measure of non-compactness β

A delicate problem Let F be an interpolation functor of exponential type of θ. Does there exist a constant C > 0 such that for any T : A → B β

• T : F(

A) → F( B)

• C β
• T : A0 → B0

1−θβ

• T : A1 → B1

θ ? This question was answered positively

◮ for the real interpolation functor F(·) = (·)θ,q by Cobos,

Fernández-Martínez and Martínez (1999), R.S. (2006),

◮ for the complex interpolation functor F(·) = [·]θ in the case

where B satisfies an approximation condition by Teixeira and Edmunds (1981), R.S. (2014).

Interpolation of entropy numbers fails

A more delicate problem Let F be an interpolation functor of exponential type of θ. Does there exist a constant C > 0 such that for any T : A → B εk0k1

• T : F(

A) → F( B)

• C εk0(T : A0 → B0)1−θ εk1(T : A1 → B1)θ ?

◮ This question was answered negatively for the real

interpolation functor F(·) = (·)θ,q by Edmunds and Netrusov (2011). The reduction

• B =

A

Recollect...

◮ The n-th entropy number εn(T) of T ∈ L(X)

εn(T) := inf

• ε > 0 ; T(UX) ⊂

n

• i=1

{xi + εUX} , xi ∈ X

• ◮ Carl’s inequality

|λn(T)| √ 2 en(T) where en(T) := ε2n−1(T)

◮ Carl-Triebel’s inequality

n

• i=1

|λi(T)| 1/n inf

k∈N k1/(2n)εk(T)

Interpolation variant of Carl-Triebel’s inequality

Theorem (2013)

Suppose that F is an interpolation functor of exponential type of θ. If T : A → A, then

• λn
• T|F(

A)

• 2en(T|A0)1−θ en(T|A1)θ

and n

• i=1
• λi
• T|F(

A)

• 1/n
• inf

k0,k1∈N (k0k1)1/2n εk0(T|A0)1−θ εk1(T|A1)θ

Generalizations of the spectral radius formula (1)

◮ Gelfand’s spectral radius formula

|λ1(T)| = lim

m→∞ T m1/m

Definition

Given T ∈ L(X), the n-th approximation number is defined by an(T) := inf

• T − S ; S ∈ L(X), rank(S) < n
• ◮ König’s (1978) formula; a generalization for higher eigenvalues

|λn(T)| = lim

m→∞ an(T m)1/m

Generalizations of the spectral radius formula (2)

◮ The n-th entropy modulus gn(T) of T ∈ L(X) is given by

n

• i=1

|λi(T)| 1/n inf

k∈N k1/(2n)εk(T) =: gn(T) ◮ Makai-Zemánek’s formula (1982)

n

• i=1

|λi(T)| 1/n = lim

m→∞ gn(T m)1/m

Problem In what form does it exist a formula for the spectral radius of T using the entropy numbers of powers of operators?

Main results - spectral entropy (1)

Theorem (2013)

Let X be a complex Banach space and T ∈ L(X). If

• λn(T)
• is

an eigenvalue sequence of T, then sup

n∈N

k−1/(2n) n

• i=1

|λi(T)| 1/n = lim

m→∞ εkm(T m)1/m

Definition

We define the k -th spectral entropy number Ek(T) by Ek(T) := lim

m→∞ εkm(T m)1/m εk(T)

Main results - spectral entropy (2)

Theorem (2013)

Fix t ∈ [1, ∞). If {tm} ⊂ N is such that lim

m→∞ t1/m m

= t, then sup

n∈N

t−1/(2n) n

• i=1

|λi(T)| 1/n = lim

m→∞ εtm(T m)1/m

Definition

Define the spectral entropy map t → Et(T) of T as follows Et(T) := lim

m→∞ εtm(T m)1/m

Main results - spectral entropy (3)

Proposition

Let X be a complex Banach space and T ∈ L(X). If

• λn(T)
• is

an eigenvalue sequence of T, then lim

m→∞ εm(T m)1/m = E1(T) = |λ1(T)|

lim

m→∞ em(T m)1/m = E2(T) = sup n∈N

n

i=1 |λi(T)|

√ 2 1/n lim

t→∞ Et(T) = ress(T)

Entropy moduli

Definition

Let ϕ: [0, ∞) → [0, ∞) be a sub-multiplicative function. Given an operator T ∈ L(X) on a complex Banach space X, we define the entropy modulus gs,ϕ(T) as follows gs,ϕ(T) := inf

k∈N k1/(2s)ϕ (εk(T)) ,

s ∈ (0, ∞)

◮ Denote by

ϕ the function on [0, ∞) given by

• ϕ(u) := lim

m→∞ ϕ(um)1/m,

u 0

◮

ϕ is sub-multiplicative and ϕ ϕ

Makai-Zemánek’s formula revisited

Theorem (2013)

Let X be an arbitrary complex Banach space and T ∈ L(X). Assume that ϕ: [0, ∞) → [0, ∞) is a nondecreasing, sub-multiplicative and right-continuous function. Then inf

t∈[1,∞) t1/(2s)

ϕ (Et(T)) = lim

m→∞ gs,ϕ(T m)1/m ,

s ∈ (0, ∞) In particular, inf

t∈[1,∞) t1/(2n)Et(T) =

n

• i=1

|λi(T)| 1/n

Interpolation of spectral entropy numbers holds

Theorem (2013)

If F be an interpolation functor of exponential type of θ, then for any T : A → A Ek0k1

• T : F(

A) → F( A)

• Ek0(T : A0 → A0)1−θ Ek1(T : A1 → A1)θ