Operators that Attain Reduced Minimum

Let H1, H2 be complex Hilbert spaces and T be a densely defined closed linear operator from its domain D(T), a dense subspace of H1, into H2. Let N(T) denote the null space of T and R(T) denote the range of T. Recall that C(T):= D(T) ∩ N(T)⊥ is called the carrier space of T and the reduced minimum modulus γ(T) of T is defined as: $${\rm{\gamma}}\left(T \right): = \inf \left\{{\left\| {T\left(x \right)} \right\|:x \in C\left(T \right),\left\| x \right\| = 1} \right\}.$$ γ ( T ) : = inf { ‖ T ( x ) ‖ : x ∈ C ( T ) , ‖ x ‖ = 1 } . Further, we say that T attains its reduced minimum modulus if there exists x0 ∈ C(T) such that ∥x0∥ = 1 and ∥T(x0)∥ = γ(T). We discuss some properties of operators that attain reduced minimum modulus. In particular, the following results are proved. 1. The operator T attains its reduced minimum modulus if and only if its Moore-Penrose inverse T† is bounded and attains its norm, that is, there exists y0 ∈ H2 such that ∥y0∥ = 1 and ∥T†∥ = ∥T†(y0)∥. 2. For each ϵ > 0, there exists a bounded operator S such that ∥S∥ ≤ ϵ and T + S attains its reduced minimum.