Banach Journal of Mathematical Analysis最新文献

In this paper, we consider a class of generalized closed linear manifolds in a nonseparable Hilbert space H, which is closely related to the generalized Fredholm theory. We first investigate properties of the set ({mathcal {B}}_{vartriangleleft }={Tin {mathcal {M}}:overline{T(H)}subset A(H)) for some (Ain {mathcal {B}}},) where ({mathcal {B}}) is a (C^*)-subalgebra of a von Neumann algebra ({mathcal {M}}). It is proved that a selfadjoint ({mathcal {B}}_{vartriangleleft }) is always an ideal in ({mathcal {M}}). In a type (textrm{II}_infty ) factor, we show that there exists a tracial weight (whose range containing infinite cardinals) such that two projections are equivalent if and only if they have the same tracial weight, which leads to a complete characterization of such selfadjoint ({mathcal {B}}_{vartriangleleft }) when ({mathcal {M}}) is a factor. Then we introduce the concept of closed manifolds with respect to a pair of C*-algebras and study some properties. Finally, when m is an infinite cardinal, as a special important case we focus on m-closed subspaces and operators which preserve m-closed subspaces. It is proved that these operators are either of rank less than m, or the generalized left semi-Fredholm operators.

Let (mathcal {B(H)}) be the collection of bounded linear operators on a complex separable Hilbert space (mathcal {H}). For (Tin mathcal {B(H)}), its numerical range and maximal numerical range are denoted by W(T) and (W_0(T)), respectively. First, we give in this paper a characterization of the maximal numerical range and, as applications, we determine maximal numerical ranges of weighted shifts, partial isometries, the Volterra integral operator and classical Toeplitz operators. Second, we study the universality of maximal numerical ranges, showing that any nonempty bounded convex closed subset of (mathbb {C}) is the maximal numerical range of some operator. Finally, we discuss the relations among the numerical range, the maximal numerical range and the spectrum. It is shown that the collection of those operators T with (W_0(T)cap sigma (T)=emptyset ) is a nonempty open subset of (mathcal {B(H)}) precisely when (dim mathcal {H}>1), and is dense precisely when (1<dim mathcal {H}<infty ). We also show that those operators T with (W_0(T)= W(T)) constitute a nowhere dense subset of (mathcal {B(H)}) precisely when (dim mathcal {H}>1)

First, we prove that a random metric space can be isometrically embedded into a complete random normed module, as an application it is easy to see that the notion of d-(sigma )-stability in a random metric space can be regarded as a special case of the notion of (sigma )-stability in a random normed module; as another application we give the final version of the characterization for a d-(sigma )-stable random metric space to be stably compact. Second, we prove that an (L^{p})-normed (L^{infty })-module is exactly generated by a complete random normed module so that the gluing property of an (L^{p})-normed (L^{infty })-module can be derived from the (sigma )-stability of the generating random normed module, as applications the direct relation between module duals and random conjugate spaces are given. Third, we prove that a random normed space is order complete iff it is ((varepsilon ,lambda ))-complete, as an application it is proved that the d-decomposability of an order complete random normed space is exactly its d-(sigma )-stability. Finally, we prove that an equivalence relation on the product space of a nonempty set X and a complete Boolean algebra B is regular iff it can be induced by a B-valued Boolean metric on X, as an application it is proved that a nonempty subset of a Boolean set (X, d) is universally complete iff it is a B-stable set defined by a regular equivalence relation.

Let G be a locally compact group and let (A_Phi (G)) be the Orlicz version of the Figà–Talamanca Herz algebra of G associated with a Young function (Phi .) We show that if (A_Phi (G)) is Arens regular, then G is discrete. We further explore the Arens regularity of (A_Phi (G)) when the underlying group G is discrete. In the running, we also show that (A_Phi (G)) is finite dimensional if and only if G is finite. Further, for amenable groups, we show that (A_Phi (G)) is reflexive if and only if G is finite, under the assumption that the associated Young function (Phi ) satisfies the MA condition.

This paper deals mainly with some aspects of the adjointable operators on Hilbert (C^*)-modules. A new tool called the generalized polar decomposition for each adjointable operator is introduced and clarified. As an application, the general theory of the weakly complementable operators is set up in the framework of Hilbert (C^*)-modules. It is proved that there exists an operator equation which has a unique solution, whereas this unique solution fails to be the reduced solution. Some investigations are also carried out in the Hilbert space case. It is proved that there exist a closed subspace M of certain Hilbert space K and an operator (Tin {mathbb {B}}(K)) such that T is (M, M)-weakly complementable, whereas T fails to be (M, M)-complementable. The solvability of the equation

is also dealt with in the Hilbert space case, where (A,Bin {mathbb {B}}(H)) are two general positive operators, and A : B denotes their parallel sum. Among other things, it is shown that there exist certain positive operators A and B on the Hilbert space (ell ^2({mathbb {N}})oplus ell ^2({mathbb {N}})) such that the above equation has no solution.

In this paper we show that if ({T_n}) is a sequence of bounded linear operators on a complex Banach space X which (nu )-converges to two different bounded linear operators T and U, then T and U have the same parts of the spectrum. In particular, we generalize the results of Sánchez-Perales and Djordjević (J Math Anal Appl 433:405–415, 2016) and of Ammar (Indag Math 28:424–435, 2017). We also investigate the spectral (nu )-continuity for the surjective spectrum.

In this note, we give a new proof of the following well-known norm formula which holds for any two orthogonal projections (P_{mathcal {T}}, P_{mathcal {S}}) on a Hilbert ({mathcal {H}},)

unless (P_{mathcal {T}}=P_{mathcal {S}}=0.) This equality was proved by Duncan and Taylor (Proc R Soc Edinb Sect A 75(2):119–129, 1975). We derive this formula from the relationship between the spectra of the sum and product of any two idempotents, as well as various norm inequalities for positive operators defined on ({mathcal {H}}.) Applications of our results are given.

By interpreting the well-known Brown–Halmos theorem for Toeplitz operators in terms of multipliers, we formulate a Brown–Halmos analogue for the product of generalized Toeplitz operators, defined as compressions of multiplication operators to closed subspaces of (L^2({mathbb {T}})). We use this to define equivalences between two operators in that class by means of multipliers between the spaces where they act. Necessary and sufficient conditions for such an equivalence to be unitary or a similarity are established. The results are applied to Toeplitz and Hankel operators, truncated Toeplitz operators, and dual truncated Toeplitz operators.

We study (ell ^r)-valued extensions of linear operators defined on Lebesgue spaces with variable exponent. Under some natural (and usual) conditions on the exponents, we characterize (1le rle infty ) such that every bounded linear operator (T:L^{q(cdot )}(Omega _2, mu )rightarrow L^{p(cdot )}(Omega _1, nu )) has a bounded (ell ^r)-valued extension. We consider both non-atomic measures and measures with atoms and show the differences that can arise. We present some applications of our results to weighted norm inequalities of linear operators and vector-valued extensions of fractional operators with rough kernel.