# Smoothness in the space of bounded linear operators on semi-Hilbert space

## TL;DR

This paper introduces and characterizes the concept of A-smoothness for bounded linear operators on semi-Hilbert spaces induced by positive operators, linking it to differentiability and analyzing specific operator classes.

## Contribution

It defines A-smoothness in semi-Hilbert spaces, provides characterizations for A-bounded and A-compact operators, and relates A-smoothness to Gâteaux differentiability.

## Key findings

- A-smoothness is characterized for A-bounded operators.
- A-smoothness of A-compact operators is analyzed via A-norm attainment.
- Gâteaux differentiability of the semi-norm is equivalent to A-smoothness.

## Abstract

Given a nonzero positive operator $A$ on a Hilbert space $\mathbb{H}$, a semi-inner product is naturally induced on $\mathbb{H}$. In this work, we introduce the notion of \emph{$A$-smoothness} for bounded linear operators on the resulting semi-Hilbert space and investigate its various properties. We provide a comprehensive characterization of the $A$-smoothness for $A$-bounded operators and further analyze the $A$-smoothness of $A$-compact operators in terms of their $A$-norm attainment sets. Utilizing these characterizations, we establish that G\^{a}teaux differentiability of the semi-norm $\|\cdot\|_A$ at an $A$-bounded operator is equivalent to its $A$-smoothness. Furthermore, we characterize the $A$-smoothness of $2\times 2$ block diagonal matrices.