The statement of the problem. In this paper we study weighted norm inequalities with matrix valued weights. Namely, let W be a d × d matrix weight, i.e. a L-function whose values are selfadjoint nonnegative d × d matrices. We suppose that the weight W is defined on the unit circle T = {z ∈ C : |z| = 1} or the real line R. Let L = L(C) be the space of all measurable vector functions on T whose C...

An explicit construction characterizing the operator-valued Bergman inner functions is given for a class of vector-valued standard weighted Bergman spaces in the unit disk. These operator-valued Bergman inner functions act as contractive multipliers from the Hardy space into the associated Bergman space, and they have a natural interpretation as transfer functions for a related class of discret...

For a Hilbert space H of functions let H H be the space of weak products of functions in H, i.e. all functions h that can be written as h = ∑∞ i=1 figi for some fi, gi ∈ H with ∑∞ i=1 ‖fi‖‖gi‖ <∞. Let D denote the Dirichlet space of the unit circle ∂D, i.e. the nontangential limits of functions f ∈ Hol(D) with ∫ D |f ′|2dA < ∞ and let Dh be the harmonic Dirichlet space, which consists of functi...

Let A be a bounded linear operator on a Banach space X. We investigate the conditions of existing rank-one operator B such that I+f(A)B is invertible for every analytic function f on sigma(A). Also we compare the invariant subspaces of f(A)B and B. This work is motivated by an operator method on the Banach space ell^2 for solving some PDEs which is extended to general operator space under some ...

let a be a bounded linear operator on a banach space x. we investigate the conditions of existing rank-one operator b such that i+f(a)b is invertible for every analytic function f on sigma(a). also we compare the invariant subspaces of f(a)b and b. this work is motivated by an operator method on the banach space ell^2 for solving some pdes which is extended to general operator space under some ...