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 ; λ I X − T is not invertible in L ( X ) ◮ The essential spectrum σ ess ( T ) := σ ( T ) T is the coset of T in the Calkin algebra L ( X ) / K ( X ) . ◮ Gelfand’s spectral radius formula m →∞ � T m � 1 / m r ( T ) := sup | λ | = lim λ ∈ σ ( T ) ◮ The essential spectral radius m →∞ � T m � 1 / m r ess ( T ) := r ( T ) = lim 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 | > r ess ( T ) � � ∞ We assign an eigenvalue sequence λ n ( T ) n = 1 for T ∈ L ( X ) from � � the elements of the set Λ( T ) ∪ r ess ( 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 | > r ess ( T ) , we let λ n ( T ) = λ n + 1 ( T ) = . . . = r ess ( T )

Entropy numbers Definition The n -th entropy number ε n ( T ) of T ∈ L ( X , Y ) is defined by � � n � ε n ( T ) := inf ε > 0 ; T ( U X ) ⊂ { y i + ε U Y } , y i ∈ Y i = 1 ◮ Entropy numbers are monotonne 0 � . . . � ε 3 ( T ) � ε 2 ( T ) � ε 1 ( T ) = � T � ◮ The measure of non-compactness β ( T ) := lim n →∞ ε n ( T )

Carl-Triebel’s inequality (1980) � � ∞ Let λ n ( T ) n = 1 be an eigenvalue sequence of T ∈ L ( X ) on a complex Banach space X . ◮ Carl’s inequality √ | λ n ( T ) | � 2 e n ( T ) where e n ( T ) := ε 2 n − 1 ( T ) ◮ Carl-Triebel’s inequality � n � 1 / n � k ∈ N k 1 / ( 2 n ) ε k ( T ) | λ i ( T ) | � inf i = 1

Banach couple ◮ We call � A := ( A 0 , A 1 ) a Banach couple if both A 0 and A 1 are Banach spaces such that A 0 , A 1 ֒ → X For a given Banach couple � A , we define spaces ◮ intersection A 0 ∩ A 1 with the norm � � � a � A 0 ∩ A 1 = max � a � A 0 , � a � A 1 ◮ sum A 0 + A 1 with the norm � � � a � A 0 + A 1 = inf � a 0 � A 0 + � a 1 � A 1 a = a 0 + a 1

Interpolation functor ◮ By T : � A → � B we denote an operator T : A 0 + A 1 → B 0 + B 1 , such that T | A j ∈ L ( A j , B j ) , j = 0 , 1 Definition By an interpolation functor we mean a mapping F : � B → B ◮ A 0 ∩ A 1 ⊂ F ( � A ) ⊂ A 0 + A 1 for any � A ∈ � B � � F ( � ⊂ F ( � A , � � B ∈ � B and T : � A → � ◮ T A ) B ) for any B

Interpolation functor of exponential type of θ For all interpolation functors F � � � T � F ( � B ) � C max � T � A 0 → B 0 , � T � A 1 → B 1 A ) →F ( � If in addition there exists θ ∈ ( 0 , 1 ) such that B ) � C � T � 1 − θ A 0 → B 0 � T � θ � T � F ( � A 1 → B 1 , A ) →F ( � 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 ε n ( T ) of T ∈ L ( X , Y ) � � n � ε n ( T ) := inf ε > 0 ; T ( U X ) ⊂ { y i + ε U Y } , y i ∈ Y i = 1 ◮ The measure of non-compactness β ( T ) := lim n →∞ ε n ( 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 � � � � 1 − θ β � � θ ? T : F ( � A ) → F ( � β B ) � C β T : A 0 → B 0 T : A 1 → B 1 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 � � � C ε k 0 ( T : A 0 → B 0 ) 1 − θ ε k 1 ( T : A 1 → B 1 ) θ ? T : F ( � A ) → F ( � ε k 0 k 1 B ) ◮ 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 ε n ( T ) of T ∈ L ( X ) � � n � ε n ( T ) := inf ε > 0 ; T ( U X ) ⊂ { x i + ε U X } , x i ∈ X i = 1 ◮ Carl’s inequality √ | λ n ( T ) | � 2 e n ( T ) where e n ( T ) := ε 2 n − 1 ( T ) ◮ Carl-Triebel’s inequality � n � 1 / n � k ∈ N k 1 / ( 2 n ) ε k ( T ) | λ i ( T ) | � inf i = 1

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 � �� � � � � � 2 e n ( T | A 0 ) 1 − θ e n ( T | A 1 ) θ � λ n T | F ( � A ) and � n � 1 / n � �� � � � � k 0 , k 1 ∈ N ( k 0 k 1 ) 1 / 2 n ε k 0 ( T | A 0 ) 1 − θ ε k 1 ( T | A 1 ) θ � λ i T | F ( � inf � � A ) i = 1

Generalizations of the spectral radius formula (1) ◮ Gelfand’s spectral radius formula m →∞ � T m � 1 / m | λ 1 ( T ) | = lim Definition Given T ∈ L ( X ) , the n -th approximation number is defined by � � a n ( T ) := inf � T − S � ; S ∈ L ( X ) , rank ( S ) < n ◮ König’s (1978) formula; a generalization for higher eigenvalues m →∞ a n ( T m ) 1 / m | λ n ( T ) | = lim

Generalizations of the spectral radius formula (2) ◮ The n -th entropy modulus g n ( T ) of T ∈ L ( X ) is given by � n � 1 / n � k ∈ N k 1 / ( 2 n ) ε k ( T ) =: g n ( T ) | λ i ( T ) | � inf i = 1 ◮ Makai-Zemánek’s formula (1982) � n � 1 / n � m →∞ g n ( T m ) 1 / m | λ i ( T ) | = lim i = 1 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 λ n ( T ) is an eigenvalue sequence of T , then � n � 1 / n � m →∞ ε k m ( T m ) 1 / m k − 1 / ( 2 n ) sup | λ i ( T ) | = lim n ∈ N i = 1 Definition We define the k -th spectral entropy number E k ( T ) by m →∞ ε k m ( T m ) 1 / m � ε k ( T ) E k ( T ) := lim

Main results - spectral entropy (2) Theorem (2013) m →∞ t 1 / m Fix t ∈ [ 1 , ∞ ) . If { t m } ⊂ N is such that lim = t , then m � n � 1 / n � m →∞ ε t m ( T m ) 1 / m t − 1 / ( 2 n ) sup | λ i ( T ) | = lim n ∈ N i = 1 Definition Define the spectral entropy map t �→ E t ( T ) of T as follows m →∞ ε t m ( T m ) 1 / m E t ( T ) := lim

Main results - spectral entropy (3) Proposition � � Let X be a complex Banach space and T ∈ L ( X ) . If λ n ( T ) is an eigenvalue sequence of T , then m →∞ ε m ( T m ) 1 / m = E 1 ( T ) = | λ 1 ( T ) | lim �� n � 1 / n i = 1 | λ i ( T ) | m →∞ e m ( T m ) 1 / m = E 2 ( T ) = sup lim √ 2 n ∈ N t →∞ E t ( T ) = r ess ( T ) lim

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 g s ,ϕ ( T ) as follows k ∈ N k 1 / ( 2 s ) ϕ ( ε k ( T )) , g s ,ϕ ( T ) := inf s ∈ ( 0 , ∞ ) ◮ Denote by � ϕ the function on [ 0 , ∞ ) given by m →∞ ϕ ( u m ) 1 / m , ϕ ( u ) := lim � 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 t ∈ [ 1 , ∞ ) t 1 / ( 2 s ) � m →∞ g s ,ϕ ( T m ) 1 / m , inf ϕ ( E t ( T )) = lim s ∈ ( 0 , ∞ ) In particular, � n � 1 / n � t ∈ [ 1 , ∞ ) t 1 / ( 2 n ) E t ( T ) = inf | λ i ( T ) | i = 1

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 � � � E k 0 ( T : A 0 → A 0 ) 1 − θ E k 1 ( T : A 1 → A 1 ) θ T : F ( � A ) → F ( � E k 0 k 1 A )