Operators that attain the reduced minimum

Abstract

Let $H_1,H_2$ be complex Hilbert spaces and $T$ be a densely defined closed linear operator from its domain $D(T)$, a dense subspace of $H_1$, into $H_2$. Let $N(T)$ denote the null space of $T$ and $R(T)$ denote the range of $T$. Recall that $C(T):=D(T)\cap N(T{)}^{\perp }$ is called the {\it carrier space of} $T$ and the {\it reduced minimum modulus } $\gamma (T)$ of $T$ is defined as: $$\gamma (T):=\inf \{\Vert T(x)\Vert :x\in C(T),\Vert x\Vert =1\}.$$ Further, we say that $T$ {\it attains its reduced minimum modulus} if there exists $x_0\in C(T)$ such that $\Vert x_0\Vert =1$ and $\Vert T(x_0)\Vert =\gamma (T)$. We discuss some properties of operators that attain reduced minimum modulus. In particular, the following results are proved.

submitted to a journal. arXiv admin note: text overlap with arXiv:1606.05736, arXiv:1609.06869. Deleted the last section from the earlier version

References in corpus (1)

• On the denseness of minimum attaining operators