Search results for " Space"

Solvability of the divergence equation implies John via Poincaré inequality

2014

Abstract Let Ω ⊂ R 2 be a bounded simply connected domain. We show that, for a fixed (every) p ∈ ( 1 , ∞ ) , the divergence equation div v = f is solvable in W 0 1 , p ( Ω ) 2 for every f ∈ L 0 p ( Ω ) , if and only if Ω is a John domain, if and only if the weighted Poincare inequality ∫ Ω | u ( x ) − u Ω | q d x ≤ C ∫ Ω | ∇ u ( x ) | q  dist  ( x , ∂ Ω ) q d x holds for some (every) q ∈ [ 1 , ∞ ) . This gives a positive answer to a question raised by Russ (2013) in the case of bounded simply connected domains. In higher dimensions similar results are proved under some additional assumptions on the domain in question.

Partial *-Algebras of Operators in a PIP-Space

2009

The family of operators on a pip-space V is endowed with two, possibly different, partial multiplications, where partial means that the multiplication is not defined for any pair A,B of elements of Op(V) but only for certain couples. The two multiplications, to be called strong and weak, give rise to two different structures that coincide in certain situations. In this chapter we will discuss first the structure of Op(V) as partial *-algebra in the sense of [AIT02] and then the possibility of representing an abstract partial *-algebra into Op(V).

Explicit expressions for Sturm-Liouville operator problems

1987

Throughout this paper H will denote a complex separable Hilbert space and L(H) denotes the algebra of all bounded linear operators on H. If T lies in L(H), its spectrum σ(T) is the set of all complex numbers z such zI–T is not invertible in L(H) and its compression spectrum σcomp(T) is the set of all complex numbers z such that the range (zI-T)(H) is not dense in H ([3, p. 240]). This paper is concerned with the Sturm–Liouville operator problemwhere λ is a complex parameter and X(t), Q, Ei, Fi for i = l,2, and t∈[0,a], are bounded operators in L(H). For the scalar case, the classical Sturm-Liouville theory yields a complete solution of the problem, see [4], and [7]. For the finite-dimension…

Spectral Asymptotics for More General Operators in One Dimension

2019

In this chapter, we generalize the results of Chap. 3. The results and the main ideas are close, but not identical, to the ones of Hager (Ann Henri Poincare 7(6):1035–1064, 2006). We will use some h-pseudodifferential machinery, see for instance Dimassi and Sjostrand (Spectral Asymptotics in the Semi-classical Limit, London Mathematical Society Lecture Note Series, vol 268. Cambridge University Press, Cambridge, 1999).

Riesz-Fischer Maps, Semi-frames and Frames in Rigged Hilbert Spaces

2021

In this note we present a review, some considerations and new results about maps with values in a distribution space and domain in a σ-finite measure space X. Namely, this is a survey about Bessel maps, frames and bases (in particular Riesz and Gel’fand bases) in a distribution space. In this setting, the Riesz-Fischer maps and semi-frames are defined and new results about them are obtained. Some examples in tempered distributions space are examined.

On the fractional integral of Weyl inL p

1994

Polaroid-Type Operators

2018

In this chapter we introduce the classes of polaroid-type operators, i.e., those operators T ∈ L(X) for which the isolated points of the spectrum σ(T) are poles of the resolvent, or the isolated points of the approximate point spectrum σap(T) are left poles of the resolvent. We also consider the class of all hereditarily polaroid operators, i.e., those operators T ∈ L(X) for which all the restrictions to closed invariant subspaces are polaroid. The class of polaroid operators, as well as the class of hereditarily polaroid operators, is very large. We shall see that every generalized scalar operator is hereditarily polaroid, and this implies that many classes of operators acting on Hilbert s…

Commutative Partial O*-Algebras

2002

This chapter is devoted to the integrability of commutative partial O*-algebras. Three notions of weak commutativity, commutativity and strong commutativity of an O*-vector space are defined and investigated. In Section 3.1, we analyze the relation between the integrability of weakly commutative O*-vector space M and the commutativity of the von Neumann algebra (M w ′ )′. In Section 3.2, we study the integrable extensions of partial O*-algebras. In Section 3.3, we describe another explicit example, namely, the partial O*-algebra M[S, T] generated by two weakly commuting symmetric operators S and T defined on a common dense domain in a Hilbert space. In particular, we investigate in detail t…

Isometries between spaces of multiple Dirichlet series

2019

Abstract In this paper we study spaces of multiple Dirichlet series and their properties. We set the ground of the theory of multiple Dirichlet series and define the spaces H ∞ ( C + k ) , k ∈ N , of convergent and bounded multiple Dirichlet series on C + k . We give a representation for these Banach spaces and prove that they are all isometrically isomorphic, independently of the dimension. The analogous result for A ( C + k ) , k ∈ N , which are the spaces of multiple Dirichlet series that are convergent on C + k and define uniformly continuous functions, is obtained.

Applications in Mathematical Physics

2009

It turns out that pip-space methods have many applications in physics, although they are seldom mentioned as such. To draw on a literary analogy, like Moliere’s Monsieur Jourdain speaking in prose without knowing so, many authors have been using pip-space language without realizing it. In particular, chains or lattices of Hilbert spaces are quite common in many fields of mathematical physics. Some of these applications will be discussed at length in this chapter. To mention a few examples: quantum mechanics, in particular singular interactions (Section 7.1.3), scattering theory (Section 7.2), quantum field theory (Section 7.3), representations of Lie groups (Section 7.4), etc.