Sufficient Conditions For Compactness Research Articles

Compactness of Sobolev-type embeddings with measures

We study compactness of embeddings of Sobolev-type spaces of arbitrary integer order into function spaces on domains in [Formula: see text] with respect to upper Ahlfors regular measures [Formula: see text], that is, Borel measures whose decay on balls is dominated by a power of their radius. Sobolev-type spaces as well as target spaces considered in this paper are built upon general rearrangement-invariant function norms. Several sufficient conditions for compactness are provided and these conditions are shown to be often also necessary, yielding sharp compactness results. It is noteworthy that the only connection between the measure [Formula: see text] and the compactness criteria is how fast the measure decays on balls. Applications to Sobolev-type spaces built upon Lorentz–Zygmund norms are also presented.

Composition Operators on de Branges–Rovnyak Spaces

We study the compactness of the composition operator on de Branges–Rovnyak spaces. Inspired by a paper by Lyubarskii–Malinnikova on model spaces, we give some necessary and some sufficient conditions for compactness. In the paper of Lyubarskii-Malinnikova, the key point is some Bernstein inequality on model spaces due to Cohn (and based on a deep inequality of Axler–Chang–Sarason involving the Hardy–Littlewood maximal function). We generalize the result of Cohn to some subspace of a de Branges–Rovnyak space (in many cases dense) and then get a sufficient condition (analogue to Lyubarskii–Malinnikova’s condition) for compactness of the composition operator on that subspace.

Compact operators with BMO symbols on multiply-connected domains

In this paper we study the problem of the boundedness and compactness of the Toeplitz operator Tϕ on L2a(Ω), where Ω is a multiply-connected domain and ϕ is not bounded. We find a necessary and sufficient condition when the symbol is BMO. For this class we also show that the vanishing at the boundary of the Berezin transform is a necessary and sufficient condition for compactness. The same characterization is shown to hold when we analyze operators which are finite sums of finite products of Toeplitz operators with unbounded symbols.

Relatively compact sets in variable-exponent Lebesgue spaces

We study totally bounded sets in variable Lebesgue spaces. The full characterization of this kind of sets is given for the case of variable Lebesgue space on metric measure spaces. Furthermore, the sufficient conditions for compactness are shown without assuming log-Holder continuity of the exponent.

Lp-boundedness of Multilinear Pseudo-differential Operators on Znand Tn

The aim of this paper is to introduce and study multilinear pseudo-differential operators on Zn and Tn = (Rn/ 2πZn) the n-torus. More precisely, we give sufficient conditions and sometimes necessary conditions for Lp-boundedness of these classes of operators. L2-boundedness results for multilinear pseudo-differential operators on Zn and Tn with L2-symbols are stated. The proofs of these results are based on elementary estimates on the multilinear Rihaczek transforms for functions in L2(Zn) respectively L2(Tn) which are also introduced. We study the weak continuity of multilinear operators on the m-fold product of Lebesgue spaces Lpj(Zn), j = 1,...,m and the link with the continuity of multilinear pseudo-differential operators on Zn. Necessary and sufficient conditions for multilinear pseudo-differential operators on Zn or Tn to be a Hilbert-Schmidt operators are also given. We give a necessary condition for a multilinear pseudo-differential operators on Zn to be compact. A sufficient condition for compactness is also given.

Abstract limit J -spaces

We investigate the limit J-spaces corresponding to the general real method. These interpolation spaces are defined by Banach sequence lattices and include those spaces that arise by the choice e = 0 in the definition of the real method. We pay especial attention to spaces generated by rearrangement-invariant sequence spaces. We establish necessary and sufficient conditions for compactness of interpolated operators between limit J-spaces. We also study the relationships between J- and K-spaces and we derive some interpolation formulae for notable couples of function spaces, couples of spaces of operators and also couples of sequence spaces.

Compact embedding in Besov spaces and B-separable elleptic operators

Necessary and sufficient conditions for compactness of sets in Banach space valued Besov class Bp,qs(Ω;E) is derived. The embedding theorems in Besov-Lions type spaces Bp,ql,s(Ω;E0, E) are studied, where E0, E are two Banach spaces and E0 ⊂ E. The most regular class of interpolation space Eα, between E0 and E are found such that the mixed differential operator Dα is bounded and compact from Bp,ql,s(Ω;E0,E) to Bp,qs(Ω;Eα) and Ehrling-Nirenberg-Gagliardo type sharp estimates established. By using these results the separability of differential operators with variable coefficients and the maximal B-regularity of parabolic Cauchy problem are obtained. In applications, the infinite systems of the elliptic partial differential equations and parabolic Cauchy problems are studied.

Compact multipliers on spaces of analytic functions

In the paper compact multiplier operators $$M_{\lambda}: X \rightarrow E$$ from Banach spaces of analytic functions on the unit disk into Banach sequence lattices are studied. If $$H_{\infty} \hookrightarrow X \hookrightarrow H_{2}$$ , then the characterization of compact multipliers is obtained through calculating the Hausdorff measure of noncompactness of diagonal operators between Banach sequence lattices. Furthermore, in the general case $$H_{\infty} \hookrightarrow X \hookrightarrow H_{1}$$ , necessary and sufficient conditions for compactness are presented.

Compact composition operators on a Hilbert space of Dirichlet series

We study the compactness of composition operators on the Hilbert space of Dirichlet series with square summable coefficients. In particular, we give some necessary and sufficient conditions for compactness. We also describe the spectrum of such operators, and we extend our work to some weighted spaces.

Linear and ellipsoidal restrictions in linear regression

The problem of combining linear and ellipsoidal restrictions in linear regression is investigated. Necessary and sufficient conditions for compactness of the restriction set are proved assuring the existence of a minimax estimator. When the restriction set is not compact a minimax estimator may still exist for special loss functions arid regression designs

Weak limits of projections and compactness of subspace lattices

A strongly closed lattice of projections on a Hilbert space is compact if the associated algebra of operators has a weakly dense subset of compact operators. If the lattice is commutative, there are necessary and sufficient conditions for compactness, one in terms of the structure of the lattice, and the other in terms of a measure on the lattice. There are many examples of compact lattices, and two main types of examples of noncompact lattices. Compactness is also related to the study of weak limits of certain projections.

Compact sets in the spaceL p (O,T; B)

A characterization of compact sets in Lp (0, T; B) is given, where 1⩽P⩾∞ and B is a Banach space. For the existence of solutions in nonlinear boundary value problems by the compactness method, the point is to obtain compactness in a space Lp (0,T; B) from estimates with values in some spaces X, Y or B where X⊂B⊂Y with compact imbedding X→B. Using the present characterization for this kind of situations, sufficient conditions for compactness are given with optimal parameters. As an example, it is proved that if {fn} is bounded in Lq(0,T; B) and in L loc 1 (0, T; X) and if {∂fn/∂t} is bounded in L loc 1 (0, T; Y) then {fn} is relatively compact in Lp(0,T; B), ∀p<q.

Separation theorems and minimax theorems for fuzzy sets

In this paper, we prove some theorems on fuzzy sets. We first show that, in order to demonstrate that the equality of shadows ofA andB implies the equality ofA andB, it is necessary to assume thatA andB are closed and thatSH(A)=SH(B) for any closed hyperplane hyperplaneH. We also obtain a separation theorem for two convex fuzzy sets in a Hilbert space. Finally, we investigate results relating to minimax theorems for fuzzy sets. We obtain a necessary and sufficient condition for compactness.

On the compactness of certain integral operators

This paper develops practical methods for deciding whether a given kernel function induces a compact integral operator from certain spaces of functions, defined on a compact subset Ω of R n , into the space of continuous functions over Ω. Necessary and sufficient conditions for compactness are introduced, and several tests for deciding if these conditions are satisfied are developed. The paper concludes with an illustration of the practical use of the theory.

Necessary and sufficient conditions for compactness of reciprocal RC two-port networks

Necessary and sufficient conditions are given for the compactness of the open-circuit impedance (or short-circuit admittance) matrix of a passive reciprocal RC two-port network when the specified short-circuit admittance (or open-circuit impedance) matrix of the network is compact. The conditions are applicable to unsymmetrical as well as to symmetrical reciprocal RC two-ports.

Compactness of the structure space of a ring

Jacobson [I ] has shown that a topology may be defined on the set SA of primitive ideals of any nonradical ring A. With this topology SA is called the structure space of A. The topology is given by defining closure: if T = { p I is a set of primitive ideals then T is the set of primitive ideals which contain nf {p IpE T}. One of Jacobson's results is that if A has a unit then SA is compact. By working with open sets rather than closure we obtain a simpler proof of this fact and also a new sufficient condition for compactness: every 2-sided ideal of A is finitely generated. Let p, q, *, be points of SA (primitive ideals). For each xEA write (x) for the principal (2 sided) ideal generated by x, and let Ux-= {I P p (X) I

Convergence of Random Processes and Limit Theorems in Probability Theory

The convergence of stochastic processes is defined in terms of the so-called “weak convergence” (w. c.) of probability measures in appropriate functional spaces (c. s. m. s.).Chapter 1. Let $\Re $ be the c.s.m.s. and v a set of all finite measures on $\Re $. The distance $L(\mu _1 ,\mu _2 )$ (that is analogous to the Levy distance) is introduced, and equivalence of L-convergence and w. c. is proved. It is shown that $V\Re = (v,L)$ is c. s. m. s. Then, the necessary and sufficient conditions for compactness in $V\Re $ are given.In section 1.6 the concept of “characteristic functionals” is applied to the study of w. cc of measures in Hilbert space.Chapter 2. On the basis of the above results the necessary and sufficient compactness conditions for families of probability measures in spaces $C[0,1]$ and $D[0,1]$ (space of functions that are continuous in $[0,1]$ except for jumps) are formulated.Chapter 3. The general form of the “invariance principle” for the sums of independent random variables is developed....