Real and complex operator norms

The numerical range of a bounded linear operator on a complex Banach space need not be convex unlike that on a Hilbert space. The aim of this paper is to study operators $T$ on $ \ell^2_p $ for which the numerical range is convex. We also obtain a nice relation between $V(T)$ and $ V(T^t)$ considering $ T \in \mathbb{L} (\ell_p^2) $ and $ T^t \in \mathbb{L} (\ell_q^2) ,$ where $T^t$ denotes the transpose of $T$ and $p$ and $q$ are conjugate real numbers i.e., $ 1 <p,q< \infty...

Let $H$ be a reflexive, dense, separable, infinite dimensional complex Hilbert space and let $B(H)$ be the algebra of all bounded linear operators on $H$. In this paper, we carry out characterizations of norm-attainable operators in normed spaces. We give conditions for norm-attainability of linear functionals in Banach spaces, non-power operators on $H$ and elementary operators. Lastly, we characterize a new notion of norm-attainability for power operators in normed spaces.

We study two classes of bounded operators on mixed norm Lebesgue spaces, namely composition operators and product operators. A complete description of bounded composition operators on mixed norm Lebesgue spaces are given. For a certain class of integral operators, we provide sufficient conditions for boundedness. We conclude by applying the developed technique to the investigation of Hardy-Steklov type operators.

We modify the very well known theory of normed spaces $(E, \norm)$ within functional analysis by considering a sequence $(\norm_n : n\in\N)$ of norms, where $\norm_n$ is defined on the product space $E^n$ for each $n\in\N$. Our theory is analogous to, but distinct from, an existing theory of `operator spaces'; it is designed to relate to general spaces $L^p$ for $p\in [1,\infty]$, and in particular to $L^1$-spaces, rather than to $L^2$-spaces. After recalling in Chapter 1...

This article introduces several new upper bounds for the $q$-numerical radius of bounded linear operators on complex Hilbert spaces. Our results refine some of the existing upper bounds in this field. The $q$-numerical radius inequalities of products and commutators of operators follow as special cases. Finally, some new inequalities for the $q$-numerical radius of $2 \times 2$ operator matrices are established.

We give a lower bound for the numerical index of the real space $L_p(\mu)$ showing, in particular, that it is non-zero for $p\neq 2$. In other words, it is shown that for every bounded linear operator $T$ on the real space $L_p(\mu)$, one has $$ \sup{\Bigl|\int |x|^{p-1}\sign(x) T x d\mu \Bigr| : x\in L_p(\mu), \|x\|=1} \geq \frac{M_p}{12\e}\|T\| $$ where $M_p=\max_{t\in[0,1]}\frac{|t^{p-1}-t|}{1+t^p}>0$ for every $p\neq 2$. It is also shown that for every bounded linear oper...

In this article, we present some new general forms of numerical radius inequalities for Hilbert space operators. The significance of these inequalities follow from the way they extend and refine some known results in this field. Among other inequalities, it is shown that if $A$ is a bounded linear operator on a complex Hilbert space, then \[{{w}^{2}}\left( A \right)\le \left\| \int_{0}^{1}{{{\left( t\left| A \right|+\left( 1-t \right)\left| {{A}^{*}} \right| \right)}^{2}}dt} ...

Let $(\Omega,\mu)$ be a $\sigma$-finite measure space. Given a Banach space $X$, let the symbol $S(X)$ stand for the unit sphere of $X$. We prove that the space $L^{\infty} (\Omega,\mu)$ of all complex-valued measurable essentially bounded functions equipped with the essential supremum norm, satisfies the Mazur-Ulam property, that is, if $X$ is any complex Banach space, every surjective isometry $\Delta: S(L^{\infty} (\Omega,\mu))\to S(X)$ admits an extension to a surjective ...

The purpose of this paper is to generalize a very famous result on products of normal operators, due to I. Kaplansky. The context of generalization is that of bounded hyponormal and unbounded normal operators on complex separable Hilbert spaces. Some examples "spice up" the paper.

A new approach to normal operators in real Hilbert spaces is discussed, and spectral decompositions are obtained, via unique associated real spectral measures on complex subsets, having a more classical flavor. The results are then applied to quaternionic normal operators, which are regarded as special classes of real normal operators.