Banach algebras defined by fractional Mikhlin-type conditions are continuously contained in Besov spaces, in such a way that the difference between the corresponding degrees of derivation can be made arbitrarily small. In this note a proof of this inclusion is given which is based on the Hadamard fractional operator and its adjoint integration operator on the positive half-line. §

We establish Lq bounds on eigenfunctions, and more generally on spectrally localized functions (spectral clusters), associated to a self-adjoint elliptic operator on a compact manifold, under the assumption that the coefficients of the operator are of regularity Cs, where 0 ≤ s ≤ 1. We also produce examples which show that these bounds are best possible for the case q =∞, and for 2 ≤ q ≤ qn.

Definitions of fractional derivatives as fractional powers of derivative operators are suggested. The Taylor series and Fourier series are used to define fractional power of self-adjoint derivative operator. The Fourier integrals and Weyl quantization procedure are applied to derive the definition of fractional derivative operator. Fractional generalization of concept of stability is considered.

This paper studies previously developed nonlinear Hilbert adjoint operator theory from a variational point of view and provides a formal justification for the use of Hamiltonian extensions via Gâteaux differentials. The primary motivation is its use in characterizing singular values of nonlinear operators, and in particular, the Hankel operator and its relationship to the state space notion of ...

It is shown that any real and even function of the phase (time) operator has a self-adjoint extension and its relation to the general phase operator problem is analyzed. Typeset using REVTEX E-mail: [email protected] E-mail: [email protected] E-mail: [email protected]

The spectral shift function for a trace class perturbation of a self-adjoint (unitary) operator plays a very important role in perturbation theory. It was introduced in a special case by I.M. Lifshitz [L] and in the general case by M.G. Krein [Kr1]. He showed that for a pair of self-adjoint (not necessarily bounded) operators A and B satisfying B −A ∈ S1 there exists a unique function ξ ∈ L 1(R...

The paper considers representing bilinear forms by linear operators in the case of a Krull valuation. More precisely, after making some suitable assumptions, we prove that if φ is a non-degenerate bilinear form, then φ is representable by a unique linear operator A whose adjoint operator A∗ exists.

We derive some rigorous results concerning the backflow operator introduced by Bracken and Melloy. We show that it is linear bounded, self adjoint, and not compact. Thus the question is underlined whether the backflow constant is an eigenvalue of the backflow operator. From the position representation of the backflow operator we obtain a more efficient method to determine the backflow constant....

Several papers relate different alternative approaches to classical concept lattices: such as property-oriented and object-oriented concept lattices and the dual concept lattices. Whereas the usual approach to the latter is via a negation operator, this paper presents a fuzzy generalization of the dual concept lattice, the dual multi-adjoint concept lattice, in which the philosophy of the multi...

The extension theory for semibounded symmetric operators is generalized by including operators acting in a triplet of Hilbert spaces. We concentrate our attention on the case where the minimal operator is essentially self-adjoint in the basic Hilbert space and construct a family of its self-adjoint extensions inside the triplet. All such extensions can be described by certain boundary condition...