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In this work we discuss several ways to extend to the context of Banach spaces the notion of Hilbert-Schmidt operators: $p$-summing operators, $\gamma$-summing or $\gamma$-radonifying operators, weakly $*1$-nuclear operators and classes of…
We consider linear operators defined on a subspace of a complex Banach space into its topological antidual acting positively in a natural sense. The goal of this paper is to investigate of this kind of operators. The main theorem is a…
We study the numerical range of bounded linear operators on quaternionic Hilbert spaces and its relation with the S-spectrum. The class of complex operators on quaternionic Hilbert spaces is introduced and the upper bild of normal complex…
In this work we provide a characterization of distinct type of (linear and non-linear) maps between Banach spaces in terms of the differentiability of certain class of Lipschitz functions. Our results are stated in an abstract bornological…
A new approach to normal operators in real Hilbert spaces is discussed, and a spectral representation is obtained, derived directly from the complex case. The results are then applied to quaternionic normal operators, regarded as a special…
We construct a general framework that generates classes of multilinear operators between Banach spaces which encompasses, as particular cases, the several classes of summing type multilinear operators that have been studied individually in…
We study some classes of algebras of operators on non-Archimedean Banach spaces. In particular, we propose a non-Archimedean version of the crossed product construction.
We first strictly expressed the basic notions and research methods of abstract operators, which systematically expounded the main results of abstract operator theory. By combining abstract operators with the Laplace transform, we can easily…
We study Birkhoff-James orthogonality of compact (bounded) linear operators between Hilbert spaces and Banach spaces. Applying the notion of semi-inner-products in normed linear spaces and some related geometric ideas, we generalize and…
We study the problem of the existence of a common algebraic complement for a pair of closed subspaces of a Banach space. We prove the following two characterizations: (1) The pairs of subspaces of a Banach space with a common complement…
In this paper, considering a real Banach sequence lattice Y instead of a Lebesgue sequence space $l_p$ we generalize p-convexity of a linear operator $T:E\to X$, where E is a Banach space and X is a Banach lattice. Then we prove that basic…
Lecture note topics: 1. Some tools from real and complex analysis, 2. Hilbert spaces, 3. Banach spaces, 4. Compact operators and their spectra, 5. Intermezzo: reproducing kernel Hilbert spaces, 6. Banach algebras ,7. Spectral theory of…
We introduce the notions of L(H)-valued norms and Banach spaces with respect to L(H)-valued norms. In particular, we introduce Hilbert spaces with respect to L(H)-valued inner products. In addition, we provide several fundamental examples…
We study Birkhoff-James orthogonality and isosceles orthogonality of bounded linear operators between Hilbert spaces and Banach spaces. We explore Birkhoff-James orthogonality of bounded linear operators in light of a new notion introduced…
We work with very general Banach spaces of analytic functions in the disk or other domains which satisfy a minimum number of natural axioms. Among the preliminary results, we discuss some implications of the basic axioms and identify all…
In the early 1970's Eisenberg and Hedlund investigated relationships between expansivity and spectrum of operators on Banach spaces. In this paper we establish relationships between notions of expansivity and hypercyclicity, supercyclicity,…
Inspired by the theories of Kaplansky-Hilbert modules and probability theory in vector lattices, we generalise functional analysis by replacing the scalars $\mathbb{R}$ or $\mathbb{C}$ by a real or complex Dedekind complete unital…
Operator representation of Cole-Hopf transform is obtained based on the logarithmic representation of infinitesimal generators. For this purpose the relativistic formulation of abstract evolution equation is introduced. Even independent of…
The article is devoted to quasilinear operators in spaces over quaternions and octonions. Spectral theory of bounded and unbounded operators is investigated. Analogs of C^* algebras are defined and studied. Among main results are analogs of…
We show that there exists a Banach space $E$ with the following properties: the Banach algebra $\mathscr{B}(E)$ of bounded, linear operators on $E$ has a singular extension which splits algebraically, but it does not split strongly, and the…