Real and complex operator norms

For any complex Banach space $X$ and each $p \in [1,\infty)$, we introduce the $p$-Bohr radius of order $N(\in \mathbb{N})$ is $\widetilde{R}_{p,N}(X)$ defined by $$ \widetilde{R}_{p,N}(X)=\sup \left\{r\geq 0: \sum_{k=0}^{N}\norm{x_k}^p r^{pk} \leq \norm{f}^p_{H^{\infty}(\mathbb{D}, X)}\right\}, $$ where $f(z)=\sum_{k=0}^{\infty} x_{k}z^k \in H^{\infty}(\mathbb{D}, X)$. Here $\mathbb{D}= \{z\in \mathbb{C}: |z| <1\}$ denotes the unit disk. We also introduce the following geome...

We introduce two kinds of operator-valued norms. One of them is an $L(H)$-valued norm. The other one is an $L(C(K))$-valued norm. We characterize the completeness with respect to a bounded $L(H)$-valued norm. Furthermore, for a given Banach space $\textbf{B}$, we provide an $L(C(K))$-valued norm on $\textbf{B}$. and we introduce an $L(C(K))$-valued norm on a Banach space satisfying special properties.

Let $H_1$ and $H_2$ be complex Hilbert spaces and $T:H_1\rightarrow H_2$ be a bounded linear operator. We say $T$ to be norm attaining, if there exists $x\in H_1$ with $\|x\|=1$ such that $\|Tx\|=\|T\|$. If for every closed subspace $M$ of $H_1$, the restriction $T|_{M}:M\rightarrow H_2$ is norm attaining then, $T$ is called absolutely norm attaining operator or $\mathcal{AN}$-operator. If we replace the norm of the operator by the minimum modulus $m(T)=\inf{\{\|Tx\|:x\in H_1...

We compute the operator $(p,q)$-norm of some $n\times n$ complex matrices, which can be seen as bounded linear operators from the $n$ dimensional Banach space $\ell^p(n)$ to $\ell^q(n)$. We have shown that a special matrix $A=\begin{pmatrix} 8 & 1 & 6 \\ 3 & 5 & 7 \\ 4 & 9 & 2 \end{pmatrix}$ which corresponds to a magic square has $\|A\|_{p,p} = \max \{\|A\xi\|_p : \xi\in\ell^p(n), \|\xi\|_p=1\} =15$ for any $p\in [1,\infty]$. In this paper, we extend this result and we compu...

The powerful concept of an operator ideal on the class of all Banach spaces makes sense in the real and in the complex case. In both settings we may, for example, consider compact, nuclear, or $2$--summing operators, where the definitions are adapted to each other in a natural way. This paper deals with the question whether or not that fact is based on a general philosophy. Does there exists a one--to--one correspondence between ``real properties'' and ``complex properties'' ...

As a cornerstone of functional analysis, Hahn Banach theorem constitutes an indispensable tool of modern analysis where its impact extends beyond the frontiers of linear functional analysis into several other domains of mathematics, including complex analysis, partial differential equations and ergodic theory besides many more. The paper is an attempt to draw attention to certain applications of the Hahn Banach theorem which are less familiar to the mathematical community, ap...

We obtain various upper bounds for the numerical radius $w(T)$ of a bounded linear operator $T$ defined on a complex Hilbert space $\mathcal{H}$, by developing the upper bounds for the $\alpha$-norm of $T$, which is defined as $\|T\|_{\alpha}= \sup \left\{ \sqrt{\alpha |\langle Tx,x \rangle|^2+ (1-\alpha)\|Tx\|^2 } : x\in \mathcal{H}, \|x\|=1 \right\}$ for $ 0\leq \alpha \leq 1 $. Further, we prove that \begin{eqnarray*} w(T) &\leq & \sqrt{\left( \min_{\alpha \in [0,1]}\left\...

We introduce a new norm on the space of bounded linear operators on a complex Hilbert space, which generalizes the numerical radius norm, the usual operator norm and the modified Davis-Wielandt radius. We study basic properties of this norm, including the upper and the lower bounds for it. As an application of the present study, we estimate bounds for the numerical radius of bounded linear operators. We illustrate with examples that our results improve on some of the importan...

Several unitarily invariant norm inequalities and numerical radius inequalities for Hilbert space operators are studied. We investigate some necessary and sufficient conditions for the parallelism of two bounded operators. For a finite rank operator $A,$ it is shown that \begin{eqnarray*} \|A\|_{p} &\leq &\left(\textit{rank} \, A\right)^{1/{2p}} \|A\|_{2p} \,\, \leq \,\, \left(\textit{rank} \, A\right)^{{(2p-1)}/{2p^2}} \|A\|_{2p^2}, \quad \textit{for all $p\geq 1 $} \end{eqn...

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 $f$-algebra $\mathbb{L}$; such an algebra can be represented as a suitable space of continuous functions. We set up the basic theory of $\mathbb{L}$-normed and $\mathbb{L}$-Banach spaces and bounded operators between them, we discuss the $\mathbb{L}...