Dilations of Banach space operator valued functions
We give universal upper bounds on the relative dimensions of isotypic components of a tensor product of representations of the linear group GL(n) and universal upper bounds on the relative dimensions of irreducible components of a tensor product of representations of the special linear group SL(n). This problem is motivated by harmonic analysis problems, and we give some applications to the theory of Beurling-Fourier algebras.
The notion of ℱ-approximate order unit norm for ordered ℱ-bimodules is introduced and characterized in terms of order-theoretic and geometric concepts. Using this notion, we characterize the inductive limit of matrix order unit spaces.
It is known that every operator on a (separable) Hilbert space is the direct integral of irreducible operators, but not every one is the direct sum of irreducible ones. We show that an operator can have either finitely or uncountably many reducing subspaces, and the former holds if and only if the operator is the direct sum of finitely many irreducible operators no two of which are unitarily equivalent. We also characterize operators T which are direct sums of irreducible operators in terms of the...
We deal with the existence of solutions of the Dirichlet problem for sublinear and superlinear partial differential inclusions considered as generalizations of the Euler-Lagrange equation for a certain integral functional without convexity assumption. We develop a duality theory and variational principles for this problem. As a consequence of the duality theory we give a numerical version of the variational principles which enables approximation of the solution for our problem.
We study the convergence properties of Dirichlet series for a bounded linear operator T in a Banach space X. For an increasing sequence of positive numbers and a sequence of functions analytic in neighborhoods of the spectrum σ(T), the Dirichlet series for is defined by D[f,μ;z](T) = ∑n=0∞ e-μnz fn(T), z∈ ℂ. Moreover, we introduce a family of summation methods called Dirichlet methods and study the ergodic properties of Dirichlet averages for T in the uniform operator topology.