Banach Journal of Mathematical Analysis最新文献

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).

We introduce a new norm (say (alpha )-norm) on ({mathcal {L}}({mathcal {X}}),) the space of all bounded linear operators defined on a normed linear space ({mathcal {X}}). We explore various properties of the (alpha )-norm. In addition, we study several equalities and inequalities of the (alpha )-norm of operators on ({mathcal {X}}.) As an application, we obtain an upper bound for the numerical radius of product of operators, which improves a well-known upper bound of the numerical radius for sectorial matrices. We present the (alpha )-norm of operators by using the extreme points of the unit ball of the corresponding spaces. Furthermore, we define a geometric constant (say (alpha )-index) associated with ({mathcal {X}}) and study properties of the (alpha )-index. In particular, we obtain the exact value of the (alpha )-index for some polyhedral spaces and complex Hilbert space. Finally, we study the (alpha )-index of (ell _p)-sum of normed linear spaces.

Motivated by new progress in the theory of ideals of Bloch maps, we introduce ((p,sigma ))-absolutely continuous Bloch maps with (pin [1,infty )) and (sigma in [0,1)) from the complex unit open disc (mathbb {D}) into a complex Banach space X. We prove a Pietsch domination/factorization theorem for such Bloch maps that provides a reformulation of some results on both absolutely continuous (multilinear) operators and Lipschitz operators. We also identify the spaces of ((p,sigma ))-absolutely continuous Bloch zero-preserving maps from (mathbb {D}) into (X^*) under a suitable norm (pi ^{mathcal {B}}_{p,sigma }) with the duals of the spaces of X-valued Bloch molecules on (mathbb {D}) equipped with the Bloch version of the ((p^*,sigma ))-Chevet–Saphar tensor norms.

It is well known that the composition operator on Hardy or Bergman space has a closed range if and only if its Nevanlinna counting function induces a reverse Carleson measure. Similar conclusion is not available on the Dirichlet space. Specifically, the reverse Carleson measure is not enough to ensure that the range of the corresponding composition operator is closed. However, under certain assumptions, we in this paper set the necessary and sufficient condition for a composition operator on the Dirichlet space to have closed range.

In this paper, we extend the results for approximation semigroups for general resolvent maps including various resolvents of maps on a general convex geodesic metric space. For our study, we introduce the notion of (general) resolvent maps which is a generalization of the resolvent maps in Lawson (J Lie Theory 33, 361–376, 2023) and then we prove several useful properties for the resolvent map and construct the approximation semigroups for resolvent maps. We also study the convergence of a proximal point like algorithm for the general resolvent map.

We give estimates for the measure of non-compactness of an operator interpolated by the limiting methods involving slowly varying functions. As applications we establish estimates for the measure of non-compactness of operators acting between Lorentz–Karamata spaces.