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A continuous frame is a family of vectors in a Hilbert space which allows reproductions of arbitrary elements by continuous superpositions. Associated to a given continuous frame we construct certain Banach spaces. Many classical function…
In this paper, we establish a suitable version of the Hahn-Banach theorem within the framework of Colombeau spaces, a class of spaces used to model generalized functions. Our approach addresses the case where maps are defined…
Let $X$ be a Banach space with a separable dual $X^{*}$. Let $Y\subset X$ be a closed subspace, and $f:Y\to\mathbb{R}$ a $C^{1}$-smooth function. Then we show there is a $C^{1}$ extension of $f$ to $X$.
We develop a general, functional calculus approach to approximation of $C_0$-semigroups on Banach spaces by bounded completely monotone functions of their generators. The approach comprises most of well-known approximation formulas, yields…
This paper provides some first steps in developing empirical process theory for functions taking values in a vector space. Our main results provide bounds on the entropy of classes of smooth functions taking values in a Hilbert space, by…
We explore the $k$-smoothness of bounded linear operators between Banach spaces, using the newly introduced notion of index of smoothness. The characterization of the $k$-smoothness of operators between Hilbert spaces follows as a direct…
The spectral theory on the $S$-spectrum was born out of the need to give quaternionic quantum mechanics (formulated by Birkhoff and von Neumann) a precise mathematical foundation. Then it turned out that this theory has important…
We consider a continuous family of linear elliptic differential operators of arbitrary order over a smooth compact manifold with boundary. Assuming constant dimension of the spaces of inner solutions, we prove that the orthogonalized…
We analyze the spectra of generalized Fibonacci and Fibonacci-like operators in Banach space $l^1$. Some of the results have application in population dynamics.
This paper presents Stieltjes-type integration for operator-valued functions with respect to spectral families. The relation between Riemann-Stieltjes integrals associated with some classes of spectral families including, in particular,…
We construct a new kind of measures, called projection families, which generalize the classical notion of vector and operator-valued measures. The maximal class of reasonable functions admits an integral with respect to a projection family,…
Let us consider a Banach space $X$ with the property that every real-valued Lipschitz function $f$ can be uniformly approximated by a Lipschitz, $C^1$-smooth function $g$ with $\Lip(g)\le C \Lip(f)$ (with $C$ depending only on the space…
The main purpose of this work is the construction of an analytic functional calculus for Clifford operators, which are operators acting on certain modules over Clifford algebras. Unlike in some preceding works by other authors, we use a…
We study ordinal-indexed, multi-layer iterations of bounded operator transforms and prove convergence to spectral/ergodic projections under functional-calculus hypotheses. For normal operators on Hilbert space and polynomial or holomorphic…
The theory of Banach spaces of Dirichlet series has drawn an increasing attention in the recent 25 years. One of the main interest of this new theory is that of defining analogues of the classical spaces of analytic functions on the unit…
This is the survey of results about norm one projections and 1-complemented subspaces in K\"othe function spaces and Banach sequence spaces. The historical development of the theory is presented from the 1930's to the newest ideas. Proofs…
This work will be centered in commutative Banach subalgebras of the algebra of bounded linear operators defined on a Free Banach spaces of countable type. The main goal of this work wil be to formulate a representation theorem for these…
We prove two theorems about differentiable functions on the Banach space C(K), where K is compact. (i) If C(K) admits a non-trivial function of class C^m and of bounded support, then all continuous real-valued functions on C(K) may be…
Classical spectral theory provides powerful tools for analyzing linear operators, but does not extend naturally to nonlinear or compositional settings. In particular, there is no general way to transport spectral invariants in a functorial…
The authors lay the foundations for the study of normal families of holomorphic functions and mappings on an infinite-dimensional normed linear space. Characterizations of normal families, in terms of value distribution, spherical…