Let A and M be closed linear operators defined on a complex Banach space X. Using operator-valued Fourier multipliers theorems, we obtain necessary and sufficient conditions to guarantee existence and uniqueness of periodic solutions to the equation d dt (Mu(t)) = Au(t) + f(t), in terms of either boundedness or R-boundedness of the modified resolvent operator determined by the equation. Our res...

In this paper we consider a general class of systems determined by operator valued measures which are assumed to be countably additive in the strong operator topology. This replaces our previous assumption of countable additivity in the uniform operator topology by the weaker assumption. Under the relaxed assumption plus an additional assumption requiring the existence of a dominating measure, ...

We give a natural definition of the Morrey spaces for Radon measures which may be non-doubling but satisfy the growth condition. In these spaces we investigate the behavior of the maximal operator, the fractional integral operator, the singular integral operator and their vector-valued extensions.

We study the stability properties of nonlinear multi-task regression in reproducing Hilbert spaces with operator-valued kernels. Such kernels, a.k.a. multi-task kernels, are appropriate for learning problems with nonscalar outputs like multi-task learning and structured output prediction. We show that multi-task kernel regression algorithms are uniformly stable in the general case of infinite-d...

In the signal-processing literature, a frame is mechanism for performing analysis and reconstruction in Hilbert space. By contrast, quantum theory, positive operator-valued measure (POVM) decomposes Hilbert-space vector purpose of computing measurement probabilities. Frames their most common generalizations can be seen to give rise POVMs, but does every reasonable POVM arise from type frame? We...

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...