On the denseness of minimum attaining operator-valued functions.

Let U be a bounded open subset of C and H be a complex separable Hilbert space. We define the following classes of functions on U ¯. C (U ¯ , B (H)) = { f : U ¯ → B (H) : f is continuous } C m (U ¯ , B (H)) = { f ∈ C (U ¯ , B (H)) : f (z) is minimum attaining for every z ∈ U } C r m (U ¯ , B (H)) = { f ∈ C (U ¯ , B (H)) : f (z) is reduced minimum attaining for every z ∈ U } C h (U ¯ , B (H)) = { f ∈ C (U ¯ , B (H)) : | f | is harmonic } Along with a few finer results on denseness of minimum attaining operators, this article primarily deals with denseness of (i) C m (U ¯ , B (H)) in C (U ¯ , B (H)) and (ii) C r m (U ¯ , B (H)) in C h (U ¯ , B (H)) with respect to the supremum norm. [ABSTRACT FROM AUTHOR]