In this paper we study the Foias-Williams operator T (Hg) = S∗ Hg 0 S where g ∈ L∞, and Hg is a Hankel operator with symbol g. We exhibit a relationship between the similarity of T (Hg) to a contraction and the rate of decay of {|gn|}n=0, the absolute values of the Fourier coefficients of the symbol g. Let H denote a complex Hilbert space and let L(H) denote the algebra of all bounded linear op...

A theory of singular and quasi-bounded temperatures associated with the heat equation is developed. Guided by the harmonic case [1] an operator S is defined on the set M of non-negative functions on an open set of Rn+l which admit superthermic majorants. Quasi-bounded and singular functions in M are defined from the operator S . The basic properties of S are discussed. Non-negative temperatures...

We show that each power bounded operator with spectral radius equal to one on a reflexive Banach space has a nonzero vector which is not supercyclic. Equivalently, the operator has a nontrivial closed invariant homogeneous subset. Moreover, the operator has a nontrivial closed invariant cone if 1 belongs to its spectrum. This generalizes the corresponding results for Hilbert space operators. Fo...

An analytic self-map φ : D → D of the open unit disk D in the complex plane induces the composition operator Cφ on H(D), the space of holomorphic functions on D, defined by Cφ(f) = f ◦φ. A basic goal in the study of composition operators is to relate function theoretic properties of φ to operator theoretic properties of Cφ. Here we review some results that characterize when φ induces a bounded ...

We analyze the limit of the spectrum of a geometric Dirac-type operator under a collapse with bounded diameter and bounded sectional curvature. In the case of a smooth limit space B, we show that the limit of the spectrum is given by the spectrum of a certain first-order differential operator on B, which can be constructed using superconnections. In the case of a general limit space X , we expr...

The operator Aε = D1g1(x1/ε, x2)D1 + D2g2(x1/ε, x2)D2 is considered in L2(R ), where gj(x1, x2), j = 1, 2, are periodic in x1 with period 1, bounded and positive definite. Let function Q(x1, x2) be bounded, positive definite and periodic in x1 with period 1. Let Q(x1, x2) = Q(x1/ε, x2). The behavior of the operator (Aε + Q ) as ε → 0 is studied. It is proved that the operator (Aε +Q ) tends to ...