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: 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. 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)∥. For each ϵ > 0, there exists a bounded operator S such that ∥S∥ ≤ ϵ and T + S attains its reduced minimum. [ABSTRACT FROM AUTHOR]