Kulkarni, S. H. and Ramesh, G.
(2020)
Operators that Attain Reduced Minimum.
Indian Journal of Pure and Applied Mathematics, 51 (4).
pp. 1615-1631.
ISSN 0019-5588
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Abstract
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.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.
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