Closedness of the k-numerical range. Issue 11 (24th July 2022)

• Record Type: Journal Article
• Authors: Chan, J. T.; Li, C. K.; Poon, Y. T.
• Is Part Of: Linear & multilinear algebra. Volume 70:Issue 11(2022)
• Page Start: 2178
• Page End: 2186
• Publication Date: 2022-07-24
• Subjects: Numerical range -- k-numerical range -- essential numerical range; 47A12; Algebras, Linear -- Periodicals; Multilinear algebra -- Periodicals; 512.505
• DOI: 10.1080/03081087.2020.1790483
• Languages: English
• ISSNs: 0308-1087

Abstract

Abstract : Let H be an infinite dimensional complex Hilbert space and let B ( H ) be the algebra of all bounded linear operators on H . For every positive integer k and A ∈ B ( H ), the k -numerical range of A is the set 1 W k ( A ) = ∑ j = 1 k ⟨ A x j, x j ⟩ : { x 1, …, x k } is an orthonormal set in H . In this note, we show that the closure of W k ( A ) can be written as the convex hull of sets involving the essential numerical range of A and W ℓ ( A ) for 1 ≤ ℓ ≤ k . We also show that if W k ( A ) is closed, then W ℓ ( A ) is also closed for 1 ≤ ℓ ≤ k .