Let Ω be an irreducible bounded symmetric domain of rank r in C d . Let K be the maximal compact subgroup of the identity component G of the biholomorphic automorphism group of the domain Ω. The group K consisting of linear transformations acts naturally on any d-tuple T = (T 1 , . . . , T d ) of commuting bounded linear operators. If the orbit of this action modulo unitary equivalence is a singleton, then we say that T is K-homogeneous. In this paper, we obtain a model for a certain class of K-homogeneous d-tuple T as the operators of multiplication by the coordinate functions z 1 , . . . , z d on a reproducing kernel Hilbert space of holomorphic functions defined on Ω. Using this model we obtain a criterion for (i) boundedness, (ii) membership in the Cowen-Douglas class, (iii) unitary equivalence and similarity of these d-tuples. In particular, we show that the adjoint of the d-tuple of multiplication by the coordinate functions on the weighted Bergman spaces are in the Cowen-Douglas class B 1 (Ω). For an irreducible bounded symmetric domain Ω of rank 2, an explicit description of the operator d i=1 T * i T i is given. In general, based on this formula, we make a conjecture giving the form of this operator.
Given a pair of positive real numbers α, β and a sesqui-analytic function K on a bounded domain Ω ⊂ C m , in this paper, we investigate the properties of the sesqui-analytic function K (α,β) , taking values in m × m matrices. One of the key findings is that K (α,β) is non-negative definite whenever K α and K β are non-negative definite. In this case, a realization of the Hilbert module determined by the kernel K (α,β) is obtained. Let Mi, i = 1, 2, be two Hilbert modules over the polynomial ring C[z1, . . . , zm]. Then C[z1, . . . , z2m] acts naturally on the tensor product M1 ⊗ M2. The restriction of this action to the polynomial ring C[z1, . . . , zm] obtained using the restriction map p → p |∆ leads to a natural decomposition of the tensor product M1 ⊗ M2, which is investigated. Two of the initial pieces in this decomposition are identified.
Let Ω be an irreducible bounded symmetric domain of rank r in C d . Let K be the maximal compact subgroup of the identity component G of the biholomorphic automorphism group of the domain Ω. The group K consisting of linear transformations acts naturally on any d-tuple T = (T1, . . . , T d ) of commuting bounded linear operators. If the orbit of this action modulo unitary equivalence is a singleton, then we say that T is K-homogeneous. In this paper, we obtain a model for all K-homogeneous d-tuple T as the operators of multiplication by the coordinate functions z1, . . . , z d on a reproducing kernel Hilbert space of holomorphic functions defined on Ω. Using this model we obtain a criterion for (i) boundedness, (ii) membership in the Cowen-Douglas class (iii) unitary equivalence and similarity of these d-tuples. In particular, we show that the adjoint of the d-tuple of multiplication by the coordinate functions on the weighted Bergman spaces are in the Cowen-Douglas class B1(Ω). For a bounded symmetric domain Ω of rank 2, an explicit description of the operator d i=1 T * i Ti is given. In general, based on this formula, we make a conjecture giving the form of this operator.
We formalize the observation that the same summability methods converge in a Banach space X and its dual X * . At the same time we determine conditions under which these methods converge in the weak and weak*-topologies on X and X * respectively. We also derive a general limitation theorem, which yields a necessary condition for the convergence of a summability method in X. These results are then illustrated by applications to a wide variety of function spaces, including spaces of continuous functions, Lebesgue spaces, the disk algebra, Hardy and Bergman spaces, the BMOA space, the Bloch space, and de Branges-Rovnyak spaces. Our approach shows that all these applications flow from just two abstract theorems.
We address the question of the analyticity of a rank one perturbation of an analytic operator. If Mz is the bounded operator of multiplication by z on a functional Hilbert space Hκ and f ∈ H with f (0) = 0, then Mz + f ⊗ 1 is always analytic. If f (0) = 0, then the analyticity of Mz + f ⊗ 1 is characterized in terms of the membership to Hκ of the formal power series obtained by multiplying f (z) byAs an application, we discuss the problem of the invariance of the left spectrum under rank one perturbation. In particular, we show that the left spectrum σ l (T + f ⊗ g) of the rank one perturbation T + f ⊗ g, g ∈ ker(T * ), of a cyclic analytic left invertible bounded linear operator T coincides with the left spectrum of T except the point f, g . In general, the point f, g may or may not belong to σ l (T + f ⊗ g). However, if it belongs to σ l (T + f ⊗ g)\{0}, then it is a simple eigenvalue of T + f ⊗ g.
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