A generalization of the 0-numerical range

Faculty of Mining, Geology and Petroleum Engineering, University of Zagreb, Pierottijeva 6, 10000 Zagreb, Croatia
e-mail: rajna.rajic@zg.hinet.hr

Abstract. Let H be a complex Hilbert space. Given a bounded linear operator A on H, we describe the set

Rⁿ(A) = {V^*AW : V,W : Cⁿ → H, V^*V = W^*W = I_n, V^*W = 0}.

It is shown that the closed matricial convex hull of Rⁿ(A) is a closed ball of radius

min {|A - λI| : λ ∈ C}

centered at the origin.

2000 Mathematics Subject Classification. 47A12.

Key words and phrases. q-numerical range of an operator.

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