Operator Algebras Generated by Left Invertibles Derek Desantis - PowerPoint PPT Presentation

Operator Algebras Generated by Left Invertibles Introduction Operator Algebras Generated by Left Invertibles Derek Desantis University of Nebraska, Lincoln GPOTS, May 2018

Operator Algebras Generated by Left Invertibles Program Outline Motivation - Frames Background A sequence { f n } in a Hilbert space H is called a frame if there exists constants 0 < A < B such that for each x ∈ H , � |� x, f n �| 2 ≤ B � x � 2 A � x � 2 ≤ n

Operator Algebras Generated by Left Invertibles Program Outline Motivation - Frames Background A sequence { f n } in a Hilbert space H is called a frame if there exists constants 0 < A < B such that for each x ∈ H , � |� x, f n �| 2 ≤ B � x � 2 A � x � 2 ≤ n We can associate to each frame { f n } a dual frame { g n } such that � x = � x, g n � f n n

Operator Algebras Generated by Left Invertibles Program Outline Motivation - Frames Background A sequence { f n } in a Hilbert space H is called a frame if there exists constants 0 < A < B such that for each x ∈ H , � |� x, f n �| 2 ≤ B � x � 2 A � x � 2 ≤ n We can associate to each frame { f n } a dual frame { g n } such that � x = � x, g n � f n n If { f k } frame for H , and T has closed range, then { Tf k } is a frame for T H .

Operator Algebras Generated by Left Invertibles Program Outline Closed Range Operators Definition Let T ∈ B ( H ) have closed range. There is a unique operator T † ∈ B ( H ) called the Moore-Penrose inverse of T such that 1 T † Tx = x for all x ∈ ker( T ) ⊥ 2 T † y = 0 for all y ∈ ( T H ) ⊥ .

Operator Algebras Generated by Left Invertibles Program Outline Closed Range Operators Definition Let T ∈ B ( H ) have closed range. There is a unique operator T † ∈ B ( H ) called the Moore-Penrose inverse of T such that 1 T † Tx = x for all x ∈ ker( T ) ⊥ 2 T † y = 0 for all y ∈ ( T H ) ⊥ . Example Let T ∈ B ( ℓ 2 ) be given by Te n = w n e n , n ≥ 0. If 0 < c < | w n | , then T is left invertible and � 0 n = 0 T † e n = w − 1 n e n − 1 n ≥ 1

Operator Algebras Generated by Left Invertibles Program Outline Closed Range Operators Definition Let T ∈ B ( H ) have closed range. There is a unique operator T † ∈ B ( H ) called the Moore-Penrose inverse of T such that 1 T † Tx = x for all x ∈ ker( T ) ⊥ 2 T † y = 0 for all y ∈ ( T H ) ⊥ . Example Let T ∈ B ( ℓ 2 ) be given by Te n = w n e n , n ≥ 0. If 0 < c < | w n | , then T is left invertible and � 0 n = 0 T † e n = w − 1 n e n − 1 n ≥ 1 If T is an isometry, then T † = T ∗ .

Operator Algebras Generated by Left Invertibles Program Outline Research Program Remark C*-algebras generated by partial isometries (graph algebras) are well studied.

Operator Algebras Generated by Left Invertibles Program Outline Research Program Remark C*-algebras generated by partial isometries (graph algebras) are well studied. E = { r, s, E 0 , E 1 } :

Operator Algebras Generated by Left Invertibles Program Outline Research Program Program Choose a closed range operator T e for each directed edge e ∈ E 1 , subject to constraints of directed graph. What is the structure of the operator algebra Alg( T e , T † e )

Operator Algebras Generated by Left Invertibles Program Outline Research Program Program Choose a closed range operator T e for each directed edge e ∈ E 1 , subject to constraints of directed graph. What is the structure of the operator algebra Alg( T e , T † e ) Remark Our focus is on representations afforded by the graph

Operator Algebras Generated by Left Invertibles The Algebra A T Definition Notation Given a left (but not right) invertible T ∈ B ( H ), let A T := Alg( T, T † )

Operator Algebras Generated by Left Invertibles The Algebra A T Definition Notation Given a left (but not right) invertible T ∈ B ( H ), let A T := Alg( T, T † ) Example If T = M z on H 2 ( T ), then A T is the classic Toeplitz algebra T = { T f + K : f ∈ C ( T ) , K ∈ K ( H 2 ( T )) }

Operator Algebras Generated by Left Invertibles The Algebra A T Definition Notation Given a left (but not right) invertible T ∈ B ( H ), let A T := Alg( T, T † ) Example If T = M z on H 2 ( T ), then A T is the classic Toeplitz algebra T = { T f + K : f ∈ C ( T ) , K ∈ K ( H 2 ( T )) } Remark General left invertibles have no Wold decomposition: �� � �� � T n ker( T ∗ ) T n H H � = ⊕ n n

Operator Algebras Generated by Left Invertibles The Algebra A T Definition Definition A left invertible operator T is called analytic if � T n H = 0 n

Operator Algebras Generated by Left Invertibles The Algebra A T Definition Definition A left invertible operator T is called analytic if � T n H = 0 n Theorem (D-) Let T be an analytic left invertible with ind ( T ) = − n for some positive integer n . Let { x i, 0 } n i =1 be an orthonormal basis for ker( T ∗ ) . Then

Operator Algebras Generated by Left Invertibles The Algebra A T Definition Definition A left invertible operator T is called analytic if � T n H = 0 n Theorem (D-) Let T be an analytic left invertible with ind ( T ) = − n for some positive integer n . Let { x i, 0 } n i =1 be an orthonormal basis for ker( T ∗ ) . Then x i,j := ( T †∗ ) j ( x i, 0 ) i = 1 , . . . n , j = 0 , 1 , . . . is a Schauder basis for H

Operator Algebras Generated by Left Invertibles Cowen-Douglas Operators Definition Definition An operator R ∈ B ( H ) is called Cowen-Douglas if there exists open subset Ω ⊂ σ ( R ) such that 1 ( R − λ ) H = H for all λ ∈ Ω 2 dim(ker( R − λ )) = n for all λ ∈ Ω. 3 � λ ∈ Ω ker( R − λ ) = H We denote this by R ∈ B n (Ω).

Operator Algebras Generated by Left Invertibles Cowen-Douglas Operators Definition Definition An operator R ∈ B ( H ) is called Cowen-Douglas if there exists open subset Ω ⊂ σ ( R ) such that 1 ( R − λ ) H = H for all λ ∈ Ω 2 dim(ker( R − λ )) = n for all λ ∈ Ω. 3 � λ ∈ Ω ker( R − λ ) = H We denote this by R ∈ B n (Ω). Theorem (D-) Let T ∈ B ( H ) be left invertible operator with ind ( T ) = − n , for n ≥ 1 . Then the following are equivalent:

Operator Algebras Generated by Left Invertibles Cowen-Douglas Operators Definition Definition An operator R ∈ B ( H ) is called Cowen-Douglas if there exists open subset Ω ⊂ σ ( R ) such that 1 ( R − λ ) H = H for all λ ∈ Ω 2 dim(ker( R − λ )) = n for all λ ∈ Ω. 3 � λ ∈ Ω ker( R − λ ) = H We denote this by R ∈ B n (Ω). Theorem (D-) Let T ∈ B ( H ) be left invertible operator with ind ( T ) = − n , for n ≥ 1 . Then the following are equivalent: 1 T is an analytic

Operator Algebras Generated by Left Invertibles Cowen-Douglas Operators Definition Definition An operator R ∈ B ( H ) is called Cowen-Douglas if there exists open subset Ω ⊂ σ ( R ) such that 1 ( R − λ ) H = H for all λ ∈ Ω 2 dim(ker( R − λ )) = n for all λ ∈ Ω. 3 � λ ∈ Ω ker( R − λ ) = H We denote this by R ∈ B n (Ω). Theorem (D-) Let T ∈ B ( H ) be left invertible operator with ind ( T ) = − n , for n ≥ 1 . Then the following are equivalent: 1 T is an analytic 2 There exists ǫ > 0 such that T ∗ ∈ B n (Ω) for Ω = { z : | z | < ǫ }

Operator Algebras Generated by Left Invertibles Cowen-Douglas Operators Definition Definition An operator R ∈ B ( H ) is called Cowen-Douglas if there exists open subset Ω ⊂ σ ( R ) such that 1 ( R − λ ) H = H for all λ ∈ Ω 2 dim(ker( R − λ )) = n for all λ ∈ Ω. 3 � λ ∈ Ω ker( R − λ ) = H We denote this by R ∈ B n (Ω). Theorem (D-) Let T ∈ B ( H ) be left invertible operator with ind ( T ) = − n , for n ≥ 1 . Then the following are equivalent: 1 T is an analytic 2 There exists ǫ > 0 such that T ∗ ∈ B n (Ω) for Ω = { z : | z | < ǫ } 3 There exists ǫ > 0 such that T † ∈ B n (Ω) for Ω = { z : | z | < ǫ }

Operator Algebras Generated by Left Invertibles Cowen-Douglas Operators Canonical Model Theorem If R ∈ B n (Ω) , then R is unitarily equivalent to M ∗ z on a RKHS H on Ω ∗ = { z : z ∈ Ω } . � of analytic functions

Operator Algebras Generated by Left Invertibles Cowen-Douglas Operators Canonical Model Theorem If R ∈ B n (Ω) , then R is unitarily equivalent to M ∗ z on a RKHS H on Ω ∗ = { z : z ∈ Ω } . � of analytic functions Analytic Model Let T be an analytic left invertible with ind( T ) = − n for some positive integer n , { x i,j } the basis associated with T †∗ , and Ω = { z : | z | < ǫ } as in previous theorem.

Operator Algebras Generated by Left Invertibles Cowen-Douglas Operators Canonical Model Theorem If R ∈ B n (Ω) , then R is unitarily equivalent to M ∗ z on a RKHS H on Ω ∗ = { z : z ∈ Ω } . � of analytic functions Analytic Model Let T be an analytic left invertible with ind( T ) = − n for some positive integer n , { x i,j } the basis associated with T †∗ , and Ω = { z : | z | < ǫ } as in previous theorem. Then for each λ ∈ Ω, n � � λ j x i,j x λ = i =1 j ≥ 0 exists in H .