An operator T on a complex Hilbert space H is called skew symmetric if T can be represented as a skew symmetric matrix relative to some orthonormal basis for H. In this note, we explore the structure of skew symmetric operators with disconnected spectra. Using the classical Riesz decomposition theorem, we give a decomposition of certain skew symmetric operators with disconnected spectra. Severa...

We review the relationship between positive operator-valued measures (POVMs) in quantum measurement theory and asymptotic morphisms in the C∗-algebra E-theory of Connes and Higson. The theory of asymptotic spectral measures, as introduced by Martinez and Trout [1], is integrally related to positive asymptotic morphisms on locally compact spaces via an asymptotic Riesz Representation Theorem. Ex...

It is shown that the Bochner-Riesz operator on Rn of negative order α is of restricted weak type in the critical points (p0, q0) and (q′ 0, p′0), where 1/q0 = 3/4 + α/2, q0 = p′0/3 for −3/2 < α < 0 in the two-dimensional case and 1/q0 = (n + 1 + 2α)/2n, q0 = (n − 1)p′0/(n + 1), for −(n + 1)/2 < α < −1/2 if n ≥ 3.

This paper is concerned with the construction of biorthogonal wavelet bases on n-dimensional cubes which provide Riesz bases for Sobolev and Besov spaces with homogeneous Dirichlet boundary conditions on part of the boundary. The essential point is that the primal and dual wavelets satisfy certain corresponding complementary boundary conditions. These results form the key ingredients of the con...

We prove weighted norm inequalities for fractional powers of elliptic operators together with their commutators with BMO functions, encompassing what is known for the classical Riesz potentials and elliptic operators with Gaussian domination by the classical heat operator. The method relies upon a good-λ method that does not use any size or smoothness estimates for the kernels. Indiana Univ. Ma...

Let T=H+iK be hyponormal and Q be a strictly monotone increasing continuous function on s(H ). We define ~ Q(T ) by ~ Q(T )=Q(H )+iK. In this paper, we show that if z is an isolated eigenvalue of ~ Q(T ), then the corresponding Riesz projection is self-adjoint. Also we introduce Xia spectrum and study the existence of an invariant subspace of an operator ~ Q(T ).

We show that the singular values m of the fractional integration operator of order > 0 acting from weighted L 2 spaces of functions deened on the interval (?1; 1) or (0; 1) may, in both cases, have asymptotics m ? or m ?=2 , as m ! 1, depending on the speciic choice of weight functions. Some questions of singular value decompositions are also investigated for Riesz potential operators in weight...

This paper studies a suitably normalized set of generalized Hermite polynomials and sets down a relevant Mehler formula, Rodrigues formula, and generalized translation operator. Weighted generalized Hermite polynomials are the eigenfunctions of a generalized Fourier transform which satisfies an F. and M. Riesz theorem on the absolute continuity of analytic measures. The Bose-like oscillator cal...

Let H be a Hilbert space and E a Banach space. In this note we present a sufficient condition for an operator R : H → E to be γ–radonifying in terms of Riesz sequences in H . This result is applied to recover a result of Lutz Weis and the second named author on the R-boundedness of resolvents, which is used to obtain a Datko-Pazy type theorem for the stochastic Cauchy problem. We also present s...