We study the subset in a unital C*-algebra composed of elements a such that $a{a}^{†}-a{a}^{†}$ is invertible, where $a^{†}$ denotes the Moore-Penrose inverse of a. A distinguished subset of this set is also investigated. Furthermore we study sequences of elements belonging to the aforementioned subsets.

We study discontinuous invertibility preserving linear mappings from a Banach algebra into the algebra of n × n matrices and give an explicit representation of such a mapping when n = 2.

Operators on function spaces of form Cɸf = f ∘ ɸ, where ɸ is a fixed map are called composition operators with symbol ɸ. We study such operators acting on the Hilbert Hardy space over the right half-plane and characterize the situations when they are invertible, Fredholm, unitary, and Hermitian. We determine the normal composition operators with inner, respectively with Möbius symbol. In select cases, we calculate their spectra, essential spectra, and numerical ranges.

Let T be a bounded linear operator on a complex Hilbert space H. In this paper we introduce a new class, denoted *, of operators satisfying $T^{*k}\left(\right|T²|-|T*|²)T^k\ge 0$ where k is a natural number, and we prove basic structural properties of these operators. Using these results, we also show that if E is the Riesz idempotent for a non-zero isolated point μ of the spectrum of T ∈ *, then E is self-adjoint and EH = ker(T-μ) = ker(T-μ)*. Some spectral properties are also presented.

We investigate isometric composition operators on the weighted Dirichlet space ${𝒟}_{\alpha }$ with standard weights ${(1-|z|}^2{)}^{\alpha }$, $\alpha >-1$. The main technique used comes from Martín and Vukotić who completely characterized the isometric composition operators on the classical Dirichlet space $𝒟$. We solve some of these but not in general. We also investigate the situation when ${𝒟}_{\alpha }$ is equipped with another equivalent norm.

We show that if T is an isometry (as metric spaces) from an open subgroup of the group of invertible elements in a unital semisimple commutative Banach algebra A onto a open subgroup of the group of invertible elements in a unital Banach algebra B, then $T{\left(1\right)}^{-1}T$ is an isometrical group isomorphism. In particular, $T{\left(1\right)}^{-1}T$ extends to an isometrical real algebra isomorphism from A onto B.

A characterization of isometries of complex Musielak-Orlicz spaces $L_{\Phi }$ is given. If $L_{\Phi }$ is not a Hilbert space and $U:L_{\Phi }\to L_{\Phi }$ is a surjective isometry, then there exist a regular set isomorphism τ from (T,Σ,μ) onto itself and a measurable function w such that U(f) = w ·(f ∘ τ) for all $f\in L_{\Phi }$. Isometries of real Nakano spaces, a particular case of Musielak-Orlicz spaces, are also studied.

We give a complete description of the structure of surjective isometries between the unitary groups of unital C*-algebras. While any surjective isometry between the unitary groups of von Neumann algebras can be extended to a real-linear Jordan *-isomorphism between the relevant von Neumann algebras, this is not the case for general unital C*-algebras. We show that the unitary groups of two C*-algebras are isomorphic as metric groups if and only if the C*-algebras are isomorphic in the sense that...

Let $X$ and $E$ be a Banach space and a real Banach lattice, respectively, and let $\Gamma$ denote an infinite set. We give concise proofs of the following results: (1) The dual space $X^{*}$ contains an isometric copy of $c_0$ iff $X^{*}$ contains an isometric copy of ${\ell }_{\infty }$, and (2) $E^{*}$ contains a lattice-isometric copy of $c_0\left(\Gamma \right)$ iff $E^{*}$ contains a lattice-isometric copy of ${\ell }_{\infty }\left(\Gamma \right)$.

We extend a theorem of Kato on similarity for sequences of projections in Hilbert spaces to the case of isomorphic Schauder decompositions in certain Banach spaces. To this end we use ℓψ-Hilbertian and ∞-Hilbertian Schauder decompositions instead of orthogonal Schauder decompositions, generalize the concept of an orthogonal Schauder decomposition to the case of Banach spaces and introduce the class of Banach spaces with Schauder-Orlicz decompositions. Furthermore, we generalize the notions of type,...