Spectra of a class of non-self-adjoint random Jacobi matrices Marko - PowerPoint PPT Presentation

Spectra of a class of non-self-adjoint random Jacobi matrices Marko Lindner, TU Chemnitz 2nd Najman Conference on Spectral Problems, Dubrovnik 15 May 2009 Marko Lindner Spectra of Random Jacobi Matrices

...with the help of... This talk is based on joint work with Simon N. Chandler-Wilde, Reading, UK Ratchanikorn Chonchaiya, Reading, UK Bernd Silbermann, Chemnitz, Germany and supported by the Marie Curie Fellowships MEIF-CT-2005-009758 and PERG02-GA-2007-224761 of the European Union. Marko Lindner Spectra of Random Jacobi Matrices

The Spaces This talk is about a class of bounded linear operators on a space of the form E = ℓ p ( Z N , X ) with p ∈ [1 , ∞ ], N ∈ N and X an arbitrary complex Banach space. Marko Lindner Spectra of Random Jacobi Matrices

The Spaces This talk is about a class of bounded linear operators on a space of the form E = ℓ p ( Z N , X ) with p ∈ [1 , ∞ ], N ∈ N and X an arbitrary complex Banach space. So u ∈ E iff u = ( u k ) k ∈ Z N where u k ∈ X for all k ∈ Z N and � � � u k � p � u � E = X , p < ∞ , p k ∈ Z N � u � E = k ∈ Z N � u k � X , sup p = ∞ . Marko Lindner Spectra of Random Jacobi Matrices

The Spaces Simplest example E = ℓ p = ℓ p ( Z , C ) , N = 1 , X = C Marko Lindner Spectra of Random Jacobi Matrices

The Spaces Simplest example E = ℓ p = ℓ p ( Z , C ) , N = 1 , X = C Slightly more sophisticated example E = L p ( R N ) ∼ = ℓ p ( Z N , L p ([0 , 1] N )) , X = L p ([0 , 1] N ) by identifying f ∈ L p ( R N ) with ( f | α +[0 , 1] N ) α ∈ Z N Marko Lindner Spectra of Random Jacobi Matrices

The Spaces Simplest example E = ℓ p = ℓ p ( Z , C ) , N = 1 , X = C Slightly more sophisticated example E = L p ( R N ) ∼ = ℓ p ( Z N , L p ([0 , 1] N )) , X = L p ([0 , 1] N ) by identifying f ∈ L p ( R N ) with ( f | α +[0 , 1] N ) α ∈ Z N Marko Lindner Spectra of Random Jacobi Matrices

The Operators L ( E ) ... the space of all bounded & linear operators E → E , K ( E ) ... the space of all compact operators E → E . With every operator A ∈ L ( E ) we will associate a matrix       . ... . . .. . . . . . . .       · · · · · ·  = a ij u j b i            . .. . . . ... . . . . . . with indices i , j ∈ Z N and operator entries a ij : X → X . For simplicity, we will restrict ourselves to band matrices. Marko Lindner Spectra of Random Jacobi Matrices

Operators: Notations A ∈ L ( E ) is called a Fredholm operator if its null-space is finite-dimensional and its range has finite co-dimension. This holds iff A + K ( E ) is invertible in L ( E ) / K ( E ). Marko Lindner Spectra of Random Jacobi Matrices

Limit Operators: Definition To study the asymptotics of the matrix entries a ij as ( i , j ) → ∞ , we introduce so-called limit operators. Definition Take a sequence h (1) , h (2) , ... ∈ Z N with | h ( n ) | → ∞ . If, for all i , j ∈ Z N , it holds that as n → ∞ , a i + h ( n ) , j + h ( n ) ⇒ b ij then B with [ B ] = ( b ij ) is called the limit operator of A with [ A ] = ( a ij ) w.r.t. the sequence h = ( h (1) , h (2) , ... ), and we write A h instead of B . Marko Lindner Spectra of Random Jacobi Matrices

Limit Operators: An Example Example: Let A be a discrete Schr¨ odinger operator   ... ...   ...   b − 1 1     1 1 b 0    ...   1  b 1   ... ... with the following potential b = ( ..., β, β, β, β , α, α, α , β, β , α , β, β , α, α, α , β, β, β, β , ... ) . ���� � �� � � �� � ���� ���� � �� � � �� � 1 4 3 2 2 3 4 Marko Lindner Spectra of Random Jacobi Matrices

Limit Operators: An Example b = ( ..., β, β, β, β , α, α, α , β, β , α , β, β , α, α, α , β, β, β, β , ... ) . ���� � �� � � �� � ���� ���� � �� � � �� � 1 4 3 2 2 3 4 Then all limit operators of A are of the form 0 1 0 1 ... ... ... ... B C B C ... ... B C B C β 1 α 1 B C B C , , B C B C B ... C B ... C B C B C 1 β 1 α B C B C @ A @ A ... ... ... ... Marko Lindner Spectra of Random Jacobi Matrices

Limit Operators: An Example b = ( ..., β, β, β, β , α, α, α , β, β , α , β, β , α, α, α , β, β, β, β , ... ) . ���� � �� � � �� � ���� ���� � �� � � �� � 1 4 3 2 2 3 4 Then all limit operators of A are of the form 0 1 0 1 ... ... ... ... B C B C ... ... B C B C β 1 α 1 B C B C , , B C B C B ... C B ... C B C B C 1 β 1 α B C B C @ A @ A ... ... ... ... 0 1 0 1 ... ... ... ... B C B C ... ... B C B C β 1 α 1 B C B C , B C B C B ... C B ... C B C B C 1 α 1 β B C B C @ A @ A ... ... ... ... Marko Lindner Spectra of Random Jacobi Matrices

Limit Operators: An Example b = ( ..., β, β, β, β , α, α, α , β, β , α , β, β , α, α, α , β, β, β, β , ... ) . ���� � �� � � �� � ���� ���� � �� � � �� � 1 4 3 2 2 3 4 Then all limit operators of A are of the form 0 1 0 1 ... ... ... ... B C B C ... ... B C B C β 1 α 1 B C B C , , B C B C B ... C B ... C B C B C 1 β 1 α B C B C @ A @ A ... ... ... ... 0 1 0 1 ... ... ... ... B C B C ... ... B C B C β 1 α 1 B C B C , B C B C B ... C B ... C B C B C 1 α 1 β B C B C @ A @ A ... ... ... ... or they are translates of the latter two matrices. Marko Lindner Spectra of Random Jacobi Matrices

Limit Operators and Random Matrices Example: Let again A be a discrete Schr¨ odinger operator   ... ...   ...   b − 1 1     1 1 b 0    ...   1  b 1   ... ... but now with ..., b − 1 , b 0 , b 1 , ... independent samples from a random variable with values in a compact set Σ ⊂ L ( X ). Then, with probability 1, b = ( ..., b − 1 , b 0 , b 1 , ... ) is a pseudoergodic sequence over Σ, by which we mean the following: Marko Lindner Spectra of Random Jacobi Matrices

Limit Operators and Random Matrices Then, with probability 1, b = ( ..., b − 1 , b 0 , b 1 , ... ) is a pseudoergodic sequence over Σ, by which we mean the following: Definition A sequence b = ( b k ) k ∈ Z is called pseudoergodic over Σ ⊂ L ( X ) if, for all ε > 0 and all finite vectors c = ( c i ) i ∈I with values c i ∈ Σ, there is a translate of b that matches c on I up to precision ε , i.e. ∃ m ∈ Z : max i ∈I � b i + m − c i � < ε. Marko Lindner Spectra of Random Jacobi Matrices

Limit Operators and Random Matrices Then, with probability 1, b = ( ..., b − 1 , b 0 , b 1 , ... ) is a pseudoergodic sequence over Σ, by which we mean the following: Definition A sequence b = ( b k ) k ∈ Z is called pseudoergodic over Σ ⊂ L ( X ) if, for all ε > 0 and all finite vectors c = ( c i ) i ∈I with values c i ∈ Σ, there is a translate of b that matches c on I up to precision ε , i.e. ∃ m ∈ Z : max i ∈I � b i + m − c i � < ε. If the potential b is pseudoergodic then every discrete Schr¨ odinger operator with a potential c = ( ..., c − 1 , c 0 , c 1 , ... ) over Σ (including A itself) is a limit operator of A – and vice versa. Marko Lindner Spectra of Random Jacobi Matrices

Limit Operators: Theorem on Fredholmness Let A ∈ L ( E ) be a band operator of the form A = B ∗ . A = invertible + locally compact with Then it is not hard to see that = ⇒ all limit operators of A are invertible . A Fredholm Marko Lindner Spectra of Random Jacobi Matrices

Limit Operators: Theorem on Fredholmness Let A ∈ L ( E ) be a band operator of the form A = B ∗ . A = invertible + locally compact with Then it is not hard to see that = ⇒ all limit operators of A are invertible . A Fredholm Under the additional condition that { a ij : i , j ∈ Z N } is relatively compact in L ( X ), in which case we call A a rich operator, we also have the reverse implication: Theorem Chandler-Wilde, ML 2007 The following are equivalent for all p ∈ [1 , ∞ ]: A is Fredholm on ℓ p ( Z , X ), all limit operators of A are invertible on ℓ p ( Z , X ), Marko Lindner Spectra of Random Jacobi Matrices

Limit Operators: Theorem on Fredholmness Let A ∈ L ( E ) be a band operator of the form A = B ∗ . A = invertible + locally compact with Then it is not hard to see that = ⇒ all limit operators of A are invertible . A Fredholm Under the additional condition that { a ij : i , j ∈ Z N } is relatively compact in L ( X ), in which case we call A a rich operator, we also have the reverse implication: Theorem Chandler-Wilde, ML 2007 The following are equivalent for all p ∈ [1 , ∞ ]: A is Fredholm on ℓ p ( Z , X ), all limit operators of A are invertible on ℓ p ( Z , X ), all limit operators of A are injective on ℓ ∞ ( Z , X ). Marko Lindner Spectra of Random Jacobi Matrices