Banach Journal of Mathematical Analysis最新文献

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.

We generalise a technique of Bhat and Skeide (J Funct Anal 269:1539–1562, 2015) to interpolate commuting families ({S_{i}}_{i in mathcal {I}}) of contractions on a Hilbert space (mathcal {H}), to commuting families ({T_{i}}_{i in mathcal {I}}) of contractive (mathcal {C}_{0})-semigroups on (L^{2}(prod _{i in mathcal {I}}mathbb {T}) otimes mathcal {H}). As an excursus, we provide applications of the interpolations to time-discretisation and the embedding problem. Applied to Parrott’s construction (1970), we then demonstrate for (d in mathbb {N}) with (d ge 3) the existence of commuting families ({T_{i}}_{i=1}^{d}) of contractive (mathcal {C}_{0})-semigroups which admit no simultaneous unitary dilation. As an application of these counter-examples, we obtain the residuality wrt.the topology of uniform (textsc {wot})-convergence on compact subsets of (mathbb {R}_{ge 0}^{d}) of non-unitarily dilatable and non-unitarily approximable d-parameter contractive (mathcal {C}_{0})-semigroups on separable infinite-dimensional Hilbert spaces for each (d ge 3). Similar results are also developed for d-tuples of commuting contractions. And by building on the counter-examples of Varopoulos-Kaijser (1973–74), a 0-1-result is obtained for the von Neumann inequality. Finally, we discuss applications to rigidity as well as the embedding problem, viz. that ‘typical’ pairs of commuting operators can be simultaneously embedded into commuting pairs of (mathcal {C}_{0})-semigroups, which extends results of Eisner (2009–2010).

We study in this paper the finite Jung constant, its interplay with Kottman’s constant and its meaning regarding the geometry of Banach spaces.

We find a new finite algorithm for evaluation of Lipschitz-free p-space norm in finite-dimensional Lipschitz-free p-spaces. We use this algorithm to deal with the problem of whether given p-metric spaces (mathcal {N}subset mathcal {M},) the canonical embedding of (mathcal {F}_p(mathcal {N})) into (mathcal {F}_p(mathcal {M})) is an isomorphism. The most significant result in this direction is that the answer is positive if (mathcal {N}subset mathcal {M}) are metric spaces.

For a given operator pair ((A,B)in (B(H),B(K))), we denote by (M_C) the operator acting on a complex infinite dimensional separable Hilbert space (Hoplus K) of the form (M_C=bigl ( {begin{matrix} A&{}C 0&{}B end{matrix}}bigr )). This paper focuses on the Fredholm complement problems of (M_C). Namely, via the operator pair (A, B), we look for an operator (Cin B(K,H)) such that (M_C) is Fredholm of finite ascent with nonzero nullity. As an application, we initiate the concept of the property (C) as a variant of Weyl’s theorem. At last, the stability of property (C) for (2times 2) upper triangular operator matrices is investigated by the virtue of the so-called entanglement spectra of the operator pair (A, B).