Density Invariance of Certain Operational Quantities of Bounded Linear Operators in Normed Spaces

Based on ideas of R.W. Cross, a simplified proof is presented of the density invariance of certain operational quantities associated with bounded linear operators in normed vector spaces. Let X , Y denote normed linear spaces and let T : X → Y be a bounded linear operator. Let I(X) denote the collection of infinite dimensional subspaces of X . For any subspace M of X , let SM = {m ∈ M : ‖m‖ = 1} and let T |M denote the restriction of T to M . We introduce the following quantities: Γ(T ) = inf M∈I(X) ‖T |M‖ ∆(T ) = sup M∈I(X) Γ(T |M ) τ(T ) = sup M∈I(X) inf m∈SM ‖Tm‖ ∇(T ) = inf M∈I(X) τ(T |M ) These quantities were originally introduced in [6], [7], and [8] for bounded linear operators. They were extended to unbounded linear operators in [1] and [2], and to linear relations in [4]. See these papers for applications of these quantities to the study of bounded and unbounded linear operators and relations. A linear subspace E of X is said to be a core of T if the graph of T |E is dense in the graph of T . A quantity f is said to be densely invariant if f(T |E) = f(T ) whenever E is a core of T . Our main result is the density invariance of the quantities Γ, ∆, τ , and ∇. For Γ, ∆, and τ , this result first appeared in [2]; for ∇, the first appearance is in [3]. The purpose of this note is to give a simplified proof. While the underlying ideas are the same as those in [2] and [3], the present proof has the following advantages: (1) the role played by the estimates is clearer, which considerably shortens the proof, and (2) the proof of the density invariance of ∆ is direct and more obviously parallel to the proof of the density invariance of ∇. Our proof is for the case of bounded operators, but the case of unbounded operators reduces to this case [2], and the case of linear relations reduces to the case of unbounded operators [3]. We require the following [5, Lemma IV.2.8(ii)]: The original version of this paper was written in the summer of 1995 while Ronald W. Cross was visiting the author at Indiana University South Bend. The author wishes to thank Prof. Cross for sharing his insights into linear operators and relations. 1