Sobolev orthogonal polynomials with respect to measures supported on subsets of the complex plane are considered. The connection between the following properties is studied: the multiplication operator M p(z) = zp(z) defined on the space P of algebraic polynomials with complex coefficients is bounded with respect to the norm defined by the Sobolev inner product, the supports of the measures are...

In this study, a high accuracy numerical method based on the spectral theory of compact operator for biharmonic eigenvalue equations on a spherical domain is developed. By employing the orthogonal spherical polynomials approximation and the spectral theory of compact operator, the error estimates of approximate eigenvalues and eigenfunctions are provided. By adopting orthogonal spherical base f...

This agrees with the definition of the spectrum in the matrix case, where the resolvent set comprises all complex numbers that are not eigenvalues. In terms of its spectrum, we will see that a compact operator behaves like a matrix, in the sense that its spectrum is the union of all of its eigenvalues and 0. We begin with the eigenspaces of a compact operator. We start with two lemmas that we w...

Transmission eigenvalues are points in the spectrum of the interior transmission operator, a coupled 2x2 system of elliptic partial differential equations, where one unknown function must satisfy two boundary conditions and the other must satisfy none. We show that the interior transmission eigenvalues are discrete and depend continuously on the contrast by proving that the interior transmissio...

Let X and Y be complete metric spaces of analytic functions over the unit disk in the complex plane. A linear operator T : X → Y is a bounded operator with respect to metric balls if T takes every metric ball in X into a metric ball in Y . We also say that T is metrically compact if it takes every metric ball in X into a relatively compact subset in Y . In this paper we will consider these prop...

Let A be a separable exact quasidiagonal C*-algebra. Suppose that ?: A L(H) is a faithful representation whose image does not contain nonzero compact operators. Then there exists a sequence .n : A L(H) of completely positive contractions such that &?(a)&.n(a)& 0 for all a # A, and the C*-algebra generated by .n(A) is finite dimensional for each n. As an application it is shown that if the C*-al...

Among all linear operators on Hilbert spaces, the compact ones (defined below) are the simplest, and most imitate the more familiar linear algebra of finite-dimensional operator theory. In addition, these are of considerable practical value and importance. We prove a spectral theorem for self-adjoint operators with minimal fuss. Thus, we do not invoke broader discussions of properties of spectr...

Let M be a compact spin manifold with a smooth action of the ntorus. Connes and Landi constructed θ-deformations Mθ of M , parameterized by n × n skew-symmetric matrices θ. The Mθ’s together with the canonical Dirac operator (D,H) on M are an isospectral deformation of M . The Dirac operator D defines a Lipschitz seminorm on C(Mθ), which defines a metric on the state space of C(Mθ). We show tha...

We consider an elliptic self-adjoint first-order differential operator acting on pairs (2-columns) of complex-valued half-densities over a connected compact three-dimensional manifold without boundary. The principal symbol of our operator is assumed to be trace-free. We study the spectral function which is the sum of squares of Euclidean norms of eigenfunctions evaluated at a given point of the...

Let (X, {wj} m j=1, {pj} m j=1) (2 ≤ m < ∞) be a contractive iterated function system (IFS), where X is a compact subset of R. It is well known that there exists a unique nonempty compact set K such that K = ⋃m j=1 wj(K). Moreover, the Ruelle operator on C(K) determined by the IFS (X, {wj} m j=1, {pj} m j=1) (2 ≤ m < ∞) has been introduced in [FL]. In the present paper, the Ruelle operators det...