The block numerical range of analytic operator functions

We introduce the block numerical range Wn(L) of an operator function L with respect to a decomposition H = H1⊕. . .⊕Hn of the underlying Hilbert space. Our main results include the spectral inclusion property and estimates of the norm of the resolvent for analytic L . They generalise, and improve, the corresponding results for the numerical range (which is the case n = 1) since the block numerical range is contained in, and may be much smaller than, the usual numerical range. We show that refinements of the decomposition entail inclusions between the corresponding block numerical ranges and that the block numerical range of the operator matrix function L contains those of its principal subminors. For the special case of operator polynomials, we investigate the boundedness of Wn(L) and we prove a Perron-Frobenius type result for the block numerical radius of monic operator polynomials with coefficients that are positive in Hilbert lattice sense.

Commuting Nonselfadjoint Operators and their Characteristic Operator-Functions

* Partially supported by Grant MM-428/94 of MESC.

Existence Results for Second Order Differential Equations with Nonlocal Conditions in Banach Spaces

This work is concerned with implicit second order abstract differential equations with nonlocal conditions. Assuming that the involved operators satisfy sonic compactness properties, we establish the existence of local mild solutions, the existence of global mild solutions and the existence of asymptotically almost periodic solutions.

An algebraic operator approach to the analysis of Gerber-Shiu functions

We introduce an algebraic operator framework to study discounted penalty functions in renewal risk models. For inter-arrival and claim size distributions with rational Laplace transform, the usual integral equation is transformed into a boundary value problem, which is solved by symbolic techniques. The factorization of the differential operator can be lifted to the level of boundary value problems, amounting to iteratively solving first-order problems. This leads to an explicit expression for the Gerber-Shiu function in terms of the penalty function.

Mean-Periodic Functions Associated with the Jacobi-Dunkl Operator on R

2000 Mathematics Subject Classification: 34K99, 44A15, 44A35, 42A75, 42A63

Applications of the Owa-Srivastava Operator to the Class of K-Uniformly Convex Functions

2000 Mathematics Subject Classification: Primary 30C45, 26A33; Secondary 33C15

On Generalized Weyl Fractional q-Integral Operator Involving Generalized Basic Hypergeometric Functions

Mathematics Subject Classification: 33D60, 33D90, 26A33

On Sandwich Theorem of Analytic Functions Involving Integral Operator

2000 Mathematics Subject Classification: 30C45

ZETA DETERMINANT AND OPERATOR DETERMINANTS

We apply techniques of zeta functions and regularized products theory to study the zeta determinant of a class of abstract operators with compact resolvent, and in particular the relation with other spectral functions.

Group theory and quasiprobability integrals of Wigner functions

The integral of the Wigner function of a quantum-mechanical system over a region or its boundary in the classical phase plane, is called a quasiprobability integral. Unlike a true probability integral, its value may lie outside the interval [0, 1]. It is characterized by a corresponding selfadjoint operator, to be called a region or contour operator as appropriate, which is determined by the characteristic function of that region or contour. The spectral problem is studied for commuting families of region and contour operators associated with concentric discs and circles of given radius a. Their respective eigenvalues are determined as functions of a, in terms of the Gauss-Laguerre polynomials. These polynomials provide a basis of vectors in a Hilbert space carrying the positive discrete series representation of the algebra su(1, 1) approximate to so(2, 1). The explicit relation between the spectra of operators associated with discs and circles with proportional radii, is given in terms of the discrete variable Meixner polynomials.

Some results on holomorphic generalized functions

In this note, we present three independent results within generalized complex analysis (in the Colombeau sense). The first of them deals with non-removable singularities; we construct a generalized function u on an open subset Omega of C(n), which is not a holomorphic generalized function on Omega but it is a holomorphic generalized function on Omega\S, where S is a hypersurface contained in Omega. The second result shows the existence of a holomorphic generalized function with prescribed values in the zero-set of a classical holomorphic function. The last result states the existence of a compactly supported solution to the (partial derivative) over bar operator.

Renormalizing the kinetic energy operator in elementary quantum mechanics

In this paper, we consider solutions to the three-dimensional Schrodinger equation of the form psi(r) = u(r)/r, where u(0) not equal 0. The expectation value of the kinetic energy operator for such wavefunctions diverges. We show that it is possible to introduce a potential energy with an expectation value that also diverges, exactly cancelling the kinetic energy divergence. This renormalization procedure produces a self-adjoint Hamiltonian. We solve some problems with this new Hamiltonian to illustrate its usefulness.

Complex variable positive definite functions

In this paper we develop an appropriate theory of positive definite functions on the complex plane from first principles and show some consequences of positive definiteness for meromorphic functions.