Academic literature on the topic 'Cowen-Douglas Classes'

In this report, after recalling the definition of the M¨obius group, we define homogeneous operators, that is, operators T with the property '(T) is unitarily equivalent to T for all ' in the M¨obius group and prove some properties of homogeneous operators. Following this, (i) we describe isometric operators which are homogeneous. (ii) we describe the homogeneous operators in the Cowen-Douglas class of rank 1. Finally, Multiplier representations which occur in the study of homogeneous operators are discussed. Following the proof of Kobayashi, the multiplier representations are shown to be irreducible.

In a foundational paper “Operators Possesing an Open Set of Eigenvalues” written several decades ago, Cowen and Douglas showed that an operator T on a Hilbert space ‘H possessing an open set Ω C of eigenvalues determines a holomorphic Hermitian vector bundle ET . One of the basic theorems they prove states that the unitary equivalence class of the operator T and the equivalence class of the holomorphic Hermitian vector bundle ET are in one to one correspondence. This correspondence appears somewhat mysterious until one detects the invariants for the vector bundle ET in the operator T and vice-versa. Fortunately, this is possible in some cases. Thus they point out that if the operator T possesses the additional property that the dimension of the eigenspace at ω is 1 for all ω Ω then the map ω ker(T - ω) admits a non-zero holomorphic section, say γ, and therefore defines a line bundle on Ω. As is well known, the curvature defined by the formula is a complete invariant for the line bundle . On the other hand, define and note that NT (ω)2 = 0. It follows that if T is unitarily equivalent to T˜, then the corresponding operators NT (ω) and NT˜(ω) are unitarily equivalent for all ω Ω. However, Cowen and Douglas prove the non-trivial converse, namely that if NT (ω) and NT˜(ω) are unitarily equivalent for all ω Ω then T and T˜ are unitarily equivalent. What does this have to do with the line bundles and .To answer this question, we must ask what is a complete invariant for the unitary equivalence class of the operator NT (ω). To find such a complete invariant we represent NT (ω) with respect to the orthonormal basis obtained from the two linearly independent vectors γ(ω),∂γ(ω) by Gram-Schmidt orthonormalization process. Then an easy computation shows that It then follows that is a complete invariant for NT (ω), ω Ω. This explains the relationship between the line bundle and the operator T in an explicit manner. Subsequently, in the paper ”Operators Possesing an Open Set of Eigenvalues”, Cowen and Douglas define a class of commuting operators possessing an open set of eigenvalues and attempt to provide similar computations as above. However, they give the details only for a pair of commuting operators. While the results of that paper remain true in the case of an arbitrary n tuple of commuting operators, it requires additional effort which we explain in this thesis.

In a foundational paper “Operators Possesing an Open Set of Eigenvalues” written several decades ago, Cowen and Douglas showed that an operator T on a Hilbert space ‘H possessing an open set Ω C of eigenvalues determines a holomorphic Hermitian vector bundle ET . One of the basic theorems they prove states that the unitary equivalence class of the operator T and the equivalence class of the holomorphic Hermitian vector bundle ET are in one to one correspondence. This correspondence appears somewhat mysterious until one detects the invariants for the vector bundle ET in the operator T and vice-versa. Fortunately, this is possible in some cases. Thus they point out that if the operator T possesses the additional property that the dimension of the eigenspace at ω is 1 for all ω Ω then the map ω ker(T - ω) admits a non-zero holomorphic section, say γ, and therefore defines a line bundle on Ω. As is well known, the curvature defined by the formula is a complete invariant for the line bundle . On the other hand, define and note that NT (ω)2 = 0. It follows that if T is unitarily equivalent to T˜, then the corresponding operators NT (ω) and NT˜(ω) are unitarily equivalent for all ω Ω. However, Cowen and Douglas prove the non-trivial converse, namely that if NT (ω) and NT˜(ω) are unitarily equivalent for all ω Ω then T and T˜ are unitarily equivalent. What does this have to do with the line bundles and .To answer this question, we must ask what is a complete invariant for the unitary equivalence class of the operator NT (ω). To find such a complete invariant we represent NT (ω) with respect to the orthonormal basis obtained from the two linearly independent vectors γ(ω),∂γ(ω) by Gram-Schmidt orthonormalization process. Then an easy computation shows that It then follows that is a complete invariant for NT (ω), ω Ω. This explains the relationship between the line bundle and the operator T in an explicit manner. Subsequently, in the paper ”Operators Possesing an Open Set of Eigenvalues”, Cowen and Douglas define a class of commuting operators possessing an open set of eigenvalues and attempt to provide similar computations as above. However, they give the details only for a pair of commuting operators. While the results of that paper remain true in the case of an arbitrary n tuple of commuting operators, it requires additional effort which we explain in this thesis.

A bounded operator T on a complex separable Hilbert space is said to be homogeneous if '(T ) is unitarily equivalent to T for all ' in M•ob, where M•ob is the M•obius group. A complete description of all homogeneous weighted shifts was obtained by Bagchi and Misra. The first examples of irreducible bi-lateral homogeneous 2-shifts were given by Koranyi. We describe all irreducible homogeneous 2-shifts up to unitary equivalence completing the list of homogeneous 2-shifts of Koranyi. After completing the list of all irreducible homogeneous 2-shifts, we show that every homogeneous operator whose associated representation is a direct sum of three copies of a Complementary series representation, is reducible. Moreover, we show that such an operator is either a direct sum of three bi-lateral weighted shifts, each of which is a homogeneous operator or a direct sum of a homogeneous bi-lateral weighted shift and an irreducible bi-lateral 2-shift. It is known that the characteristic function T of a homogeneous contraction T with an associated representation is of the form T (a) = L( a) T (0) R( a); where L and R are projective representations of the M•obius group M•ob with a common multiplier. We give another proof of the \product formula". We point out that the defect operators of a homogeneous contraction in B2(D) are not always quasi-invertible (recall that an operator T is said to be quasi-invertible if T is injective and ran(T ) is dense). We prove that when the defect operators of a homogeneous contraction in B2(D) are not quasi-invertible, the projective representations L and R are unitarily equivalent to the holomorphic Discrete series representations D+ 1 and D++3, respectively. Also, we prove that, when the defect operators of a homogeneous contraction in B2(D) are quasi-invertible, the two representations L and R are unitarily equivalent to certain known pairs of representations D 1; 2 and D +1; 1 ; respectively. These are described explicitly. Let G be either (i) the direct product of n-copies of the bi-holomorphic automorphism group of the disc or (ii) the bi-holomorphic automorphism group of the polydisc Dn: A commuting tuple of bounded operators T = (T1; T2; : : : ; Tn) is said to be homogeneous with respect to G if the joint spectrum of T lies in Dn and '(T); defined using the usual functional calculus, is unitarily equivalent to T for all ' 2 G: We show that a commuting tuple T in the Cowen-Douglas class of rank 1 is homogeneous with respect to G if and only if it is unitarily equivalent to the tuple of the multiplication operators on either the reproducing kernel Hilbert space with reproducing kernel n 1 i=1 (1 ziwi) i or Q n i i n; are positive real numbers, according asQG is as in (i) or 1 ; where ; i, 1 i i=1 (1 z w ) (ii). Finally, we show that a commuting tuple (T1; T2; : : : ; Tn) in the Cowen-Douglas class of rank 2 is homogeneous with respect to M•obn if and only if it is unitarily equivalent to the tuple of the multiplication operators on the reproducing kernel Hilbert space whose reproducing kernel is a product of n 1 rank one kernels and a rank two kernel. We also show that there is no irreducible tuple of operators in B2(Dn), which is homogeneous with respect to the group Aut(Dn):

A bounded operator T on a complex separable Hilbert space is said to be homogeneous if '(T ) is unitarily equivalent to T for all ' in M•ob, where M•ob is the M•obius group. A complete description of all homogeneous weighted shifts was obtained by Bagchi and Misra. The first examples of irreducible bi-lateral homogeneous 2-shifts were given by Koranyi. We describe all irreducible homogeneous 2-shifts up to unitary equivalence completing the list of homogeneous 2-shifts of Koranyi. After completing the list of all irreducible homogeneous 2-shifts, we show that every homogeneous operator whose associated representation is a direct sum of three copies of a Complementary series representation, is reducible. Moreover, we show that such an operator is either a direct sum of three bi-lateral weighted shifts, each of which is a homogeneous operator or a direct sum of a homogeneous bi-lateral weighted shift and an irreducible bi-lateral 2-shift. It is known that the characteristic function T of a homogeneous contraction T with an associated representation is of the form T (a) = L( a) T (0) R( a); where L and R are projective representations of the M•obius group M•ob with a common multiplier. We give another proof of the \product formula". We point out that the defect operators of a homogeneous contraction in B2(D) are not always quasi-invertible (recall that an operator T is said to be quasi-invertible if T is injective and ran(T ) is dense). We prove that when the defect operators of a homogeneous contraction in B2(D) are not quasi-invertible, the projective representations L and R are unitarily equivalent to the holomorphic Discrete series representations D+ 1 and D++3, respectively. Also, we prove that, when the defect operators of a homogeneous contraction in B2(D) are quasi-invertible, the two representations L and R are unitarily equivalent to certain known pairs of representations D 1; 2 and D +1; 1 ; respectively. These are described explicitly. Let G be either (i) the direct product of n-copies of the bi-holomorphic automorphism group of the disc or (ii) the bi-holomorphic automorphism group of the polydisc Dn: A commuting tuple of bounded operators T = (T1; T2; : : : ; Tn) is said to be homogeneous with respect to G if the joint spectrum of T lies in Dn and '(T); defined using the usual functional calculus, is unitarily equivalent to T for all ' 2 G: We show that a commuting tuple T in the Cowen-Douglas class of rank 1 is homogeneous with respect to G if and only if it is unitarily equivalent to the tuple of the multiplication operators on either the reproducing kernel Hilbert space with reproducing kernel n 1 i=1 (1 ziwi) i or Q n i i n; are positive real numbers, according asQG is as in (i) or 1 ; where ; i, 1 i i=1 (1 z w ) (ii). Finally, we show that a commuting tuple (T1; T2; : : : ; Tn) in the Cowen-Douglas class of rank 2 is homogeneous with respect to M•obn if and only if it is unitarily equivalent to the tuple of the multiplication operators on the reproducing kernel Hilbert space whose reproducing kernel is a product of n 1 rank one kernels and a rank two kernel. We also show that there is no irreducible tuple of operators in B2(Dn), which is homogeneous with respect to the group Aut(Dn):

The curvature of a contraction T in the Cowen-Douglas class is bounded above by the curvature of the backward shift operator. However, in general, an operator satisfying the curvature inequality need not be contractive. In this thesis, we characterize a slightly smaller class of contractions using a stronger form of the curvature inequality. Along the way, we find conditions on the metric of the holomorphic Hermitian vector bundle E corresponding to the operator T in the Cowen-Douglas class which ensures negative definiteness of the curvature function. We obtain a generalization for commuting tuples of operators in the Cowen-Douglas class. Secondly, we obtain an explicit formula for the curvature of the jet bundle of the Hermitian holomorphic bundle E f on a planar domain Ω. Here Ef is assumed to be a pull-back of the tautological bundle on gr(n, H ) by a nondegenerate holomorphic map f :Ω →Gr (n, H ). Clearly, finding relationships amongs the complex geometric invariants inherent in the short exact sequence 0 → Jk(Ef ) → Jk+1(Ef ) →J k+1(Ef )/ Jk(Ef ) → 0 is an important problem, whereJk(Ef ) represents the k-th order jet bundle. It is known that the Chern classes of these bundles must satisfy c(Jk+1(Ef )) = c(Jk(Ef )) c(Jk+1(Ef )/ Jk(Ef )). We obtain a refinement of this formula: trace Idnxn ( KJk(Ef )) - trace Idnxn ( KJk-1(Ef ))= KJk(Ef )/ Jk-1(Ef )(z).

The curvature of a contraction T in the Cowen-Douglas class is bounded above by the curvature of the backward shift operator. However, in general, an operator satisfying the curvature inequality need not be contractive. In this thesis, we characterize a slightly smaller class of contractions using a stronger form of the curvature inequality. Along the way, we find conditions on the metric of the holomorphic Hermitian vector bundle E corresponding to the operator T in the Cowen-Douglas class which ensures negative definiteness of the curvature function. We obtain a generalization for commuting tuples of operators in the Cowen-Douglas class. Secondly, we obtain an explicit formula for the curvature of the jet bundle of the Hermitian holomorphic bundle E f on a planar domain Ω. Here Ef is assumed to be a pull-back of the tautological bundle on gr(n, H ) by a nondegenerate holomorphic map f :Ω →Gr (n, H ). Clearly, finding relationships amongs the complex geometric invariants inherent in the short exact sequence 0 → Jk(Ef ) → Jk+1(Ef ) →J k+1(Ef )/ Jk(Ef ) → 0 is an important problem, whereJk(Ef ) represents the k-th order jet bundle. It is known that the Chern classes of these bundles must satisfy c(Jk+1(Ef )) = c(Jk(Ef )) c(Jk+1(Ef )/ Jk(Ef )). We obtain a refinement of this formula: trace Idnxn ( KJk(Ef )) - trace Idnxn ( KJk-1(Ef ))= KJk(Ef )/ Jk-1(Ef )(z).