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

Operator Algebras Generated by Left Invertibles Background and General Program General Program Recall Unitaries preserve orthonormal bases

Operator Algebras Generated by Left Invertibles Background and General Program General Program Recall Unitaries preserve orthonormal bases Partial isometries preserve orthonormal sets

Operator Algebras Generated by Left Invertibles Background and General Program General Program Recall Unitaries preserve orthonormal bases Partial isometries preserve orthonormal sets The adjoint of a partial isometry “walks backwards”

Operator Algebras Generated by Left Invertibles Background and General Program General Program Recall Unitaries preserve orthonormal bases Partial isometries preserve orthonormal sets The adjoint of a partial isometry “walks backwards” Remark Invertible operators preserve property of being a frame

Operator Algebras Generated by Left Invertibles Background and General Program General Program Recall Unitaries preserve orthonormal bases Partial isometries preserve orthonormal sets The adjoint of a partial isometry “walks backwards” Remark Invertible operators preserve property of being a frame Closed range operators (ran( T ) = ran( T )) preserve frames for closed subspaces

Operator Algebras Generated by Left Invertibles Background and General Program General Program Recall Unitaries preserve orthonormal bases Partial isometries preserve orthonormal sets The adjoint of a partial isometry “walks backwards” Remark Invertible operators preserve property of being a frame Closed range operators (ran( T ) = ran( T )) preserve frames for closed subspaces Question What is the analog of the adjoint for a closed range operator?

Operator Algebras Generated by Left Invertibles Background and General Program General Program 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 Background and General Program General Program 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 If T is an isometry, then T † = T ∗ .

Operator Algebras Generated by Left Invertibles Background and General Program General Program 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 If T is an isometry, then T † = T ∗ . Let T ∈ B ( ℓ 2 ) be given by Te n = w n e n +1 , n ≥ 1. If 0 < c < | w n | , then T has closed range (left invertible) and � 0 n = 1 T † e n = w − 1 n e n − 1 n ≥ 2

Operator Algebras Generated by Left Invertibles Background and General Program General Program Program For each edge e in Γ, pick operators { T e } e ∈ E 1 with closed range subject to constraints of graph. Analyze the structure of the operator algebra A Γ := Alg( { T e , T † e } e ∈ E 1 ) .

Operator Algebras Generated by Left Invertibles Background and General Program General Program Program For each edge e in Γ, pick operators { T e } e ∈ E 1 with closed range subject to constraints of graph. Analyze the structure of the operator algebra A Γ := Alg( { T e , T † e } e ∈ E 1 ) . Remark Our focus is on representations afforded by the graph

Operator Algebras Generated by Left Invertibles Background and General Program General Program Focus Let T be a left invertible operator, and T † its Moore-Penrose inverse. Set A T := Alg( T, T † ) .

Operator Algebras Generated by Left Invertibles Background and General Program General Program Focus Let T be a left invertible operator, and T † its Moore-Penrose inverse. Set A T := Alg( T, T † ) . Question 1 In what way does A T look like the C*-algebra generated by an isometry?

Operator Algebras Generated by Left Invertibles Background and General Program General Program Focus Let T be a left invertible operator, and T † its Moore-Penrose inverse. Set A T := Alg( T, T † ) . Question 1 In what way does A T look like the C*-algebra generated by an isometry? 2 What are the isomorphism classes of A T ?

Operator Algebras Generated by Left Invertibles Isometries and The Toeplitz Algebra 1 Background and General Program Basic Elements of Functional Analysis General Program 2 Isometries and The Toeplitz Algebra Decomposition of Isometries A Better Representation 3 Left Invertible Operators and Cowen-Douglas Operators Analytic Left Invertible Cowen-Douglas Operators 4 Examples and Classification Compact Operators and the Structure of A T Examples from Subnormal Operators Classification for dim ker( T ∗ ) = 1 5 Future Work

Operator Algebras Generated by Left Invertibles Isometries and The Toeplitz Algebra Decomposition of Isometries Proposition (Wold-Decomposition) If V ∈ B ( H ) is an isometry, then V = U ⊕ ( ⊕ α ∈ A S ) where U is a unitary and S is the shift operator. Namely,     V n ker( V ∗ )  �  � V n H  ⊕ H =  n ≥ 0 n ≥ 0 and | A | = dim(ker( V ∗ )) .

Operator Algebras Generated by Left Invertibles Isometries and The Toeplitz Algebra Decomposition of Isometries Proposition (Wold-Decomposition) If V ∈ B ( H ) is an isometry, then V = U ⊕ ( ⊕ α ∈ A S ) where U is a unitary and S is the shift operator. Namely,     V n ker( V ∗ )  �  � V n H  ⊕ H =  n ≥ 0 n ≥ 0 and | A | = dim(ker( V ∗ )) . Idea If one wants to analyze C ∗ ( V ) for some isometry V , one needs to understand C ∗ ( S ).

Operator Algebras Generated by Left Invertibles Isometries and The Toeplitz Algebra A Better Representation The functions e n ( z ) := z n for n ∈ Z form an orthonormal basis for L 2 ( T ) with normalized Lebesgue measure.

Operator Algebras Generated by Left Invertibles Isometries and The Toeplitz Algebra A Better Representation The functions e n ( z ) := z n for n ∈ Z form an orthonormal basis for L 2 ( T ) with normalized Lebesgue measure. Definition The Hardy Space H 2 ( T ) is subspace given by H 2 ( T ) := span { e n : n ≥ 0 } .

Operator Algebras Generated by Left Invertibles Isometries and The Toeplitz Algebra A Better Representation The functions e n ( z ) := z n for n ∈ Z form an orthonormal basis for L 2 ( T ) with normalized Lebesgue measure. Definition The Hardy Space H 2 ( T ) is subspace given by H 2 ( T ) := span { e n : n ≥ 0 } . Definition If f ∈ L ∞ ( T ), define M f ∈ B ( L 2 ( T )) via ∀ g ∈ L 2 ( T ) . M f ( g ) = fg

Operator Algebras Generated by Left Invertibles Isometries and The Toeplitz Algebra A Better Representation The functions e n ( z ) := z n for n ∈ Z form an orthonormal basis for L 2 ( T ) with normalized Lebesgue measure. Definition The Hardy Space H 2 ( T ) is subspace given by H 2 ( T ) := span { e n : n ≥ 0 } . Definition If f ∈ L ∞ ( T ), define M f ∈ B ( L 2 ( T )) via ∀ g ∈ L 2 ( T ) . M f ( g ) = fg The Toeplitz operator T f ∈ B ( H 2 ( T )) is T f := P H 2 ( T ) M f | H 2 ( T ) .

Operator Algebras Generated by Left Invertibles Isometries and The Toeplitz Algebra A Better Representation Remark The shift S ∈ B ( ℓ 2 ( N )) is unitarily equivalent to T z ∈ B ( H 2 ( T )).

Operator Algebras Generated by Left Invertibles Isometries and The Toeplitz Algebra A Better Representation Remark The shift S ∈ B ( ℓ 2 ( N )) is unitarily equivalent to T z ∈ B ( H 2 ( T )). Hence, C ∗ ( S ) ∼ = C ∗ ( T z ).

Operator Algebras Generated by Left Invertibles Isometries and The Toeplitz Algebra A Better Representation Remark The shift S ∈ B ( ℓ 2 ( N )) is unitarily equivalent to T z ∈ B ( H 2 ( T )). Hence, C ∗ ( S ) ∼ = C ∗ ( T z ). Theorem (Coburn) We have C ∗ ( T z ) = { T f + K : f ∈ C ( T ) , K ∈ K ( H 2 ( T )) } . Moreover, if A ∈ C ∗ ( T z ) , A = T f + K for exactly one f ∈ C ( T ) and K ∈ K ( H 2 ( T )) .

Operator Algebras Generated by Left Invertibles Isometries and The Toeplitz Algebra A Better Representation Remark The shift S ∈ B ( ℓ 2 ( N )) is unitarily equivalent to T z ∈ B ( H 2 ( T )). Hence, C ∗ ( S ) ∼ = C ∗ ( T z ). Theorem (Coburn) We have C ∗ ( T z ) = { T f + K : f ∈ C ( T ) , K ∈ K ( H 2 ( T )) } . Moreover, if A ∈ C ∗ ( T z ) , A = T f + K for exactly one f ∈ C ( T ) and K ∈ K ( H 2 ( T )) . Further, K ( H 2 ( T )) is the unique minimal ideal of C ∗ ( T z ) .

Operator Algebras Generated by Left Invertibles Isometries and The Toeplitz Algebra A Better Representation Remark The shift S ∈ B ( ℓ 2 ( N )) is unitarily equivalent to T z ∈ B ( H 2 ( T )). Hence, C ∗ ( S ) ∼ = C ∗ ( T z ). Theorem (Coburn) We have C ∗ ( T z ) = { T f + K : f ∈ C ( T ) , K ∈ K ( H 2 ( T )) } . Moreover, if A ∈ C ∗ ( T z ) , A = T f + K for exactly one f ∈ C ( T ) and K ∈ K ( H 2 ( T )) . Further, K ( H 2 ( T )) is the unique minimal ideal of C ∗ ( T z ) . Also I − SS ∗ , I − S ∗ S ∈ K ( H ) ,

Operator Algebras Generated by Left Invertibles Isometries and The Toeplitz Algebra A Better Representation Remark The shift S ∈ B ( ℓ 2 ( N )) is unitarily equivalent to T z ∈ B ( H 2 ( T )). Hence, C ∗ ( S ) ∼ = C ∗ ( T z ). Theorem (Coburn) We have C ∗ ( T z ) = { T f + K : f ∈ C ( T ) , K ∈ K ( H 2 ( T )) } . Moreover, if A ∈ C ∗ ( T z ) , A = T f + K for exactly one f ∈ C ( T ) and K ∈ K ( H 2 ( T )) . Further, K ( H 2 ( T )) is the unique minimal ideal of C ∗ ( T z ) . Also I − SS ∗ , I − S ∗ S ∈ K ( H ) , yielding ι π K ( H 2 ( T )) C ∗ ( T z ) 0 C ( T ) 0

Operator Algebras Generated by Left Invertibles Left Invertible Operators and Cowen-Douglas Operators 1 Background and General Program Basic Elements of Functional Analysis General Program 2 Isometries and The Toeplitz Algebra Decomposition of Isometries A Better Representation 3 Left Invertible Operators and Cowen-Douglas Operators Analytic Left Invertible Cowen-Douglas Operators 4 Examples and Classification Compact Operators and the Structure of A T Examples from Subnormal Operators Classification for dim ker( T ∗ ) = 1 5 Future Work

Operator Algebras Generated by Left Invertibles Left Invertible Operators and Cowen-Douglas Operators Analytic Left Invertible Remark General left invertibles have no Wold decomposition: �� � �� � T n ker( T ∗ ) T n H H � = ⊕ n n

Operator Algebras Generated by Left Invertibles Left Invertible Operators and Cowen-Douglas Operators Analytic Left Invertible Remark General left invertibles have no Wold decomposition: �� � �� � T n ker( T ∗ ) T n H H � = ⊕ n n Example Let H = ℓ 2 ( N ) ⊕ ℓ 2 ( Z ), and define T ∈ B ( H ) as � S � 0 T = ι U U is the bilateral shift on ℓ 2 ( Z ) and ι is inclusion.

Operator Algebras Generated by Left Invertibles Left Invertible Operators and Cowen-Douglas Operators Analytic Left Invertible Remark General left invertibles have no Wold decomposition: �� � �� � T n ker( T ∗ ) T n H H � = ⊕ n n Example Let H = ℓ 2 ( N ) ⊕ ℓ 2 ( Z ), and define T ∈ B ( H ) as � S � 0 T = ι U U is the bilateral shift on ℓ 2 ( Z ) and ι is inclusion. Definition A left invertible operator T is called analytic if � T n H = 0 . n

Operator Algebras Generated by Left Invertibles Left Invertible Operators and Cowen-Douglas Operators Analytic Left Invertible Remark If V is an analytic isometry ( U = 0 in Wold-decomposition), dim ker( V ∗ ) = n and { e i, 0 } n i =1 is an orthonormal basis for ker( V ∗ ), then e i,j = V j ( e i, 0 ) i = 1 , . . . n , j = 0 , 1 , . . . is an orthonormal basis for H .

Operator Algebras Generated by Left Invertibles Left Invertible Operators and Cowen-Douglas Operators Analytic Left Invertible Remark If V is an analytic isometry ( U = 0 in Wold-decomposition), dim ker( V ∗ ) = n and { e i, 0 } n i =1 is an orthonormal basis for ker( V ∗ ), then e i,j = V j ( e i, 0 ) i = 1 , . . . n , j = 0 , 1 , . . . is an orthonormal basis for H . Theorem (D-) Let T be an analytic left invertible with dim ker( 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 Left Invertible Operators and Cowen-Douglas Operators Analytic Left Invertible Remark If V is an analytic isometry ( U = 0 in Wold-decomposition), dim ker( V ∗ ) = n and { e i, 0 } n i =1 is an orthonormal basis for ker( V ∗ ), then e i,j = V j ( e i, 0 ) i = 1 , . . . n , j = 0 , 1 , . . . is an orthonormal basis for H . Theorem (D-) Let T be an analytic left invertible with dim ker( 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 Left Invertible Operators and Cowen-Douglas Operators Cowen-Douglas Operators Definition Given Ω ⊂ C open, n ∈ N , we say that R is Cowen-Douglas , and write R ∈ B n (Ω) if 1 Ω ⊂ σ ( R ) = { λ ⊂ C : R − λ not invertible } 2 ( R − λ ) H = H for all λ ∈ Ω 3 dim(ker( R − λ )) = n for all λ ∈ Ω. 4 � λ ∈ Ω ker( R − λ ) = H

Operator Algebras Generated by Left Invertibles Left Invertible Operators and Cowen-Douglas Operators Cowen-Douglas Operators Definition Given Ω ⊂ C open, n ∈ N , we say that R is Cowen-Douglas , and write R ∈ B n (Ω) if 1 Ω ⊂ σ ( R ) = { λ ⊂ C : R − λ not invertible } 2 ( R − λ ) H = H for all λ ∈ Ω 3 dim(ker( R − λ )) = n for all λ ∈ Ω. 4 � λ ∈ Ω ker( R − λ ) = H Theorem (D-) Let T ∈ B ( H ) be left invertible operator with dim ker( T ∗ ) = n , for n ≥ 1 . Then the following are equivalent:

Operator Algebras Generated by Left Invertibles Left Invertible Operators and Cowen-Douglas Operators Cowen-Douglas Operators Definition Given Ω ⊂ C open, n ∈ N , we say that R is Cowen-Douglas , and write R ∈ B n (Ω) if 1 Ω ⊂ σ ( R ) = { λ ⊂ C : R − λ not invertible } 2 ( R − λ ) H = H for all λ ∈ Ω 3 dim(ker( R − λ )) = n for all λ ∈ Ω. 4 � λ ∈ Ω ker( R − λ ) = H Theorem (D-) Let T ∈ B ( H ) be left invertible operator with dim ker( T ∗ ) = n , for n ≥ 1 . Then the following are equivalent: 1 T is an analytic

Operator Algebras Generated by Left Invertibles Left Invertible Operators and Cowen-Douglas Operators Cowen-Douglas Operators Definition Given Ω ⊂ C open, n ∈ N , we say that R is Cowen-Douglas , and write R ∈ B n (Ω) if 1 Ω ⊂ σ ( R ) = { λ ⊂ C : R − λ not invertible } 2 ( R − λ ) H = H for all λ ∈ Ω 3 dim(ker( R − λ )) = n for all λ ∈ Ω. 4 � λ ∈ Ω ker( R − λ ) = H Theorem (D-) Let T ∈ B ( H ) be left invertible operator with dim ker( 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 Left Invertible Operators and Cowen-Douglas Operators Cowen-Douglas Operators Definition Given Ω ⊂ C open, n ∈ N , we say that R is Cowen-Douglas , and write R ∈ B n (Ω) if 1 Ω ⊂ σ ( R ) = { λ ⊂ C : R − λ not invertible } 2 ( R − λ ) H = H for all λ ∈ Ω 3 dim(ker( R − λ )) = n for all λ ∈ Ω. 4 � λ ∈ Ω ker( R − λ ) = H Theorem (D-) Let T ∈ B ( H ) be left invertible operator with dim ker( 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 Left Invertible Operators and Cowen-Douglas Operators Cowen-Douglas Operators Analytic Model If R ∈ B n (Ω), there is a analytic map γ : Ω → H such that γ ( λ ) ∈ ker( R − λ ).

Operator Algebras Generated by Left Invertibles Left Invertible Operators and Cowen-Douglas Operators Cowen-Douglas Operators Analytic Model If R ∈ B n (Ω), there is a analytic map γ : Ω → H such that γ ( λ ) ∈ ker( R − λ ). For each f ∈ H , define a holomorphic function ˆ f over Ω ∗ := { z : z ∈ Ω } via ˆ f ( λ ) = � f, γ ( λ ) � .

Operator Algebras Generated by Left Invertibles Left Invertible Operators and Cowen-Douglas Operators Cowen-Douglas Operators Analytic Model If R ∈ B n (Ω), there is a analytic map γ : Ω → H such that γ ( λ ) ∈ ker( R − λ ). For each f ∈ H , define a holomorphic function ˆ f over Ω ∗ := { z : z ∈ Ω } via ˆ f ( λ ) = � f, γ ( λ ) � . H = { ˆ ” f : f ∈ H } . Equip with � ˆ Let f, ˆ g � = � f, g � .

Operator Algebras Generated by Left Invertibles Left Invertible Operators and Cowen-Douglas Operators Cowen-Douglas Operators Analytic Model If R ∈ B n (Ω), there is a analytic map γ : Ω → H such that γ ( λ ) ∈ ker( R − λ ). For each f ∈ H , define a holomorphic function ˆ f over Ω ∗ := { z : z ∈ Ω } via ˆ f ( λ ) = � f, γ ( λ ) � . H = { ˆ ” f : f ∈ H } . Equip with � ˆ Let f, ˆ g � = � f, g � . Then U : H → ” H via Uf = ˆ f is unitary, and ( UTf )( λ ) = � Tf, γ ( λ ) � = � f, λγ ( λ ) � = ( M z Uf )( λ )

Operator Algebras Generated by Left Invertibles Left Invertible Operators and Cowen-Douglas Operators Cowen-Douglas Operators Corollary T is unitarily equivalent to M z on a RKHS of analytic ” functions H .

Operator Algebras Generated by Left Invertibles Left Invertible Operators and Cowen-Douglas Operators Cowen-Douglas Operators Corollary T is unitarily equivalent to M z on a RKHS of analytic ” functions H . Under this identification, T † becomes “division by z ”.

Operator Algebras Generated by Left Invertibles Left Invertible Operators and Cowen-Douglas Operators Cowen-Douglas Operators Corollary T is unitarily equivalent to M z on a RKHS of analytic ” functions H . Under this identification, T † becomes “division by z ”. Lemma If T ∈ B ( H ) is left invertible with dim ker( T ∗ ) = n , then � N M � β m T † m : F is finite rank α n T n + Alg ( T, T † ) = � � F + . n =0 m =1

Operator Algebras Generated by Left Invertibles Left Invertible Operators and Cowen-Douglas Operators Cowen-Douglas Operators Corollary T is unitarily equivalent to M z on a RKHS of analytic ” functions H . Under this identification, T † becomes “division by z ”. Lemma If T ∈ B ( H ) is left invertible with dim ker( T ∗ ) = n , then � N M � β m T † m : F is finite rank α n T n + Alg ( T, T † ) = � � F + . n =0 m =1 Heuristic A T is compact perturbations of of multiplication operators with symbols Laurent series centered at zero.

Operator Algebras Generated by Left Invertibles Examples and Classification 1 Background and General Program Basic Elements of Functional Analysis General Program 2 Isometries and The Toeplitz Algebra Decomposition of Isometries A Better Representation 3 Left Invertible Operators and Cowen-Douglas Operators Analytic Left Invertible Cowen-Douglas Operators 4 Examples and Classification Compact Operators and the Structure of A T Examples from Subnormal Operators Classification for dim ker( T ∗ ) = 1 5 Future Work

Operator Algebras Generated by Left Invertibles Examples and Classification Compact Operators and the Structure of A T Theorem (D-) If T is an analytic left invertible with dim ker( T ∗ ) = 1 , then A T contains the compact operators K ( H ) . Moreover, K ( H ) is a minimal ideal of A T .

Operator Algebras Generated by Left Invertibles Examples and Classification Compact Operators and the Structure of A T Theorem (D-) If T is an analytic left invertible with dim ker( T ∗ ) = 1 , then A T contains the compact operators K ( H ) . Moreover, K ( H ) is a minimal ideal of A T . Corollary I − TT † , I − T † T ∈ K ( H ) . Thus, π ( T ) − 1 = π ( T † ) .

Operator Algebras Generated by Left Invertibles Examples and Classification Compact Operators and the Structure of A T Theorem (D-) If T is an analytic left invertible with dim ker( T ∗ ) = 1 , then A T contains the compact operators K ( H ) . Moreover, K ( H ) is a minimal ideal of A T . Corollary I − TT † , I − T † T ∈ K ( H ) . Thus, π ( T ) − 1 = π ( T † ) . Hence, we have the following: ι π 0 K ( H ) A T B 0 where B = Alg { π ( T ) , π ( T † ) } .

Operator Algebras Generated by Left Invertibles Examples and Classification Examples from Subnormal Operators Definition An operator N ∈ B ( H ) is normal if NN ∗ = N ∗ N .

Operator Algebras Generated by Left Invertibles Examples and Classification Examples from Subnormal Operators Definition An operator N ∈ B ( H ) is normal if NN ∗ = N ∗ N . An operator S ∈ B ( H ) is essentially normal if π ( S ) is normal in B ( H ) / K ( H ).

Operator Algebras Generated by Left Invertibles Examples and Classification Examples from Subnormal Operators Definition An operator N ∈ B ( H ) is normal if NN ∗ = N ∗ N . An operator S ∈ B ( H ) is essentially normal if π ( S ) is normal in B ( H ) / K ( H ). An operator S ∈ B ( H ) is subnormal if it has a normal extension: � S � A N = ∈ B ( K ) 0 B

Operator Algebras Generated by Left Invertibles Examples and Classification Examples from Subnormal Operators Definition An operator N ∈ B ( H ) is normal if NN ∗ = N ∗ N . An operator S ∈ B ( H ) is essentially normal if π ( S ) is normal in B ( H ) / K ( H ). An operator S ∈ B ( H ) is subnormal if it has a normal extension: � S � A N = ∈ B ( K ) 0 B The operator N is said to be a minimal normal extension if K has no proper subspace reducing N and containing H .

Operator Algebras Generated by Left Invertibles Examples and Classification Examples from Subnormal Operators Definition Let µ be a scalar-valued spectral measure associated to N , and f ∈ L ∞ ( σ ( N ) , µ ).

Operator Algebras Generated by Left Invertibles Examples and Classification Examples from Subnormal Operators Definition Let µ be a scalar-valued spectral measure associated to N , and f ∈ L ∞ ( σ ( N ) , µ ). Define T f ∈ B ( H ) via T f := P ( f ( N )) | H where P is the orthogonal projection of K onto H .

Operator Algebras Generated by Left Invertibles Examples and Classification Examples from Subnormal Operators Definition Let µ be a scalar-valued spectral measure associated to N , and f ∈ L ∞ ( σ ( N ) , µ ). Define T f ∈ B ( H ) via T f := P ( f ( N )) | H where P is the orthogonal projection of K onto H . Theorem (Keough, Olin and Thomson ) If S is an irreducible, subnormal, essentially normal operator, such that σ ( N ) = σ e ( S ) . Then C ∗ ( S ) = { T f + K : f ∈ C ( σ e ( S )) , K ∈ K ( H ) } . Moreover, then each element has A ∈ C ∗ ( S ) has a unique representation of the form T f + K .

Operator Algebras Generated by Left Invertibles Examples and Classification Examples from Subnormal Operators Theorem (D-) Let S be an analytic left invertible, dim ker( S ∗ ) = 1 , essentially normal, subnormal operator with N := mne ( S ) such that σ ( N ) = σ e ( S ) .

Operator Algebras Generated by Left Invertibles Examples and Classification Examples from Subnormal Operators Theorem (D-) Let S be an analytic left invertible, dim ker( S ∗ ) = 1 , essentially normal, subnormal operator with N := mne ( S ) such that σ ( N ) = σ e ( S ) . Set B = Alg { z, z − 1 } on σ e ( S ) . Then

Operator Algebras Generated by Left Invertibles Examples and Classification Examples from Subnormal Operators Theorem (D-) Let S be an analytic left invertible, dim ker( S ∗ ) = 1 , essentially normal, subnormal operator with N := mne ( S ) such that σ ( N ) = σ e ( S ) . Set B = Alg { z, z − 1 } on σ e ( S ) . Then A S = { T f + K : f ∈ B , K ∈ K ( H ) } Moreover, the representation of each element as T f + K is unique.

Operator Algebras Generated by Left Invertibles Examples and Classification Classification for dim ker( T ∗ ) = 1 Theorem (D-) Let T i , i = 1 , 2 be left invertible (analytic, dim ker( T ∗ i ) = 1 ) with A i := A T i . Suppose that φ : A 1 → A 2 is a bounded isomorphism.

Operator Algebras Generated by Left Invertibles Examples and Classification Classification for dim ker( T ∗ ) = 1 Theorem (D-) Let T i , i = 1 , 2 be left invertible (analytic, dim ker( T ∗ i ) = 1 ) with A i := A T i . Suppose that φ : A 1 → A 2 is a bounded isomorphism. Then φ = Ad V for some invertible V ∈ B ( H ) . That is, for all A ∈ A 1 , φ ( A ) = V AV − 1

Operator Algebras Generated by Left Invertibles Examples and Classification Classification for dim ker( T ∗ ) = 1 Theorem (D-) Let T i , i = 1 , 2 be left invertible (analytic, dim ker( T ∗ i ) = 1 ) with A i := A T i . Suppose that φ : A 1 → A 2 is a bounded isomorphism. Then φ = Ad V for some invertible V ∈ B ( H ) . That is, for all A ∈ A 1 , φ ( A ) = V AV − 1 Remark To distinguish these algebras by isomorphism classes, we need to classify the similarity orbit: S ( T ) := { V TV − 1 : V ∈ B ( H ) is invertible }

Operator Algebras Generated by Left Invertibles Examples and Classification Classification for dim ker( T ∗ ) = 1 Remark To determine S ( T ), suffices to identify S ( T ∗ ).

Operator Algebras Generated by Left Invertibles Examples and Classification Classification for dim ker( T ∗ ) = 1 Remark To determine S ( T ), suffices to identify S ( T ∗ ). Recall that T ∗ ∈ B 1 (Ω) for some disc Ω centered at the origin.

Operator Algebras Generated by Left Invertibles Examples and Classification Classification for dim ker( T ∗ ) = 1 Remark To determine S ( T ), suffices to identify S ( T ∗ ). Recall that T ∗ ∈ B 1 (Ω) for some disc Ω centered at the origin. Determining the similarity orbit of Cowen-Douglas operators is a classic problem.

Operator Algebras Generated by Left Invertibles Examples and Classification Classification for dim ker( T ∗ ) = 1 Remark To determine S ( T ), suffices to identify S ( T ∗ ). Recall that T ∗ ∈ B 1 (Ω) for some disc Ω centered at the origin. Determining the similarity orbit of Cowen-Douglas operators is a classic problem. Theorem (Jiang, Wang, Guo, Ji) Let A, B ∈ B 1 (Ω) . Then A is similar to B if and only if K 0 ( { A ⊕ B } ′ ) ∼ = Z

Operator Algebras Generated by Left Invertibles Future Work 1 Background and General Program Basic Elements of Functional Analysis General Program 2 Isometries and The Toeplitz Algebra Decomposition of Isometries A Better Representation 3 Left Invertible Operators and Cowen-Douglas Operators Analytic Left Invertible Cowen-Douglas Operators 4 Examples and Classification Compact Operators and the Structure of A T Examples from Subnormal Operators Classification for dim ker( T ∗ ) = 1 5 Future Work

Operator Algebras Generated by Left Invertibles Future Work Future Work: Determine the isomorphism classes for dim ker( T ∗ ) > 1.

Operator Algebras Generated by Left Invertibles Future Work Future Work: Determine the isomorphism classes for dim ker( T ∗ ) > 1. Is Rad( A T / K ( H )) = 0?

Operator Algebras Generated by Left Invertibles Future Work Future Work: Determine the isomorphism classes for dim ker( T ∗ ) > 1. Is Rad( A T / K ( H )) = 0? Any hope for non-analytic left invertibles?

Operator Algebras Generated by Left Invertibles Future Work Future Work: Determine the isomorphism classes for dim ker( T ∗ ) > 1. Is Rad( A T / K ( H )) = 0? Any hope for non-analytic left invertibles? The closure of the similarity orbit of T , S ( T ) can be expressed by spectral, Fredholm, and algebraic properties of T . If S ( T 1 ) = S ( T 2 ), is A T 1 ∼ = A T 2 ?