# zbMATH — the first resource for mathematics

A note on the boundary of the joint numerical range. (English) Zbl 1429.15019
Let $$A_{1},\dots,A_{n}$$ be self-adjoint operators on a complex Hilbert space $$H$$. The joint numerical range of the $$n$$-tuple $$A=(A_{1},\dots,A_{n})$$ is the set $$W(A)=\{(\left\langle A_{1}x,x\right\rangle\dots,\left\langle A_{n}x,x\right\rangle ):x\in H,\left\Vert x\right\Vert =1\}$$.
The author gives results about the boundary of $$W(A)$$ and its closure. It is shown that $$W(A)$$ is closed if and only if $$W_{e}(A)\subseteq W(A)$$ and every point in $$W_{e}(A)$$ is a star centre of $$W(A)$$, where $$W_{e}(A)$$ is the joint essential numerical range of $$A$$.

##### MSC:
 15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory 47A12 Numerical range, numerical radius
Full Text:
##### References:
 [1] Lancaster, JS, The boundary of the numerical range, Proc Am Math Soc, 49, 393-398, (1975) · Zbl 0306.47001 [2] Li, CK; Poon, YT, The joint essential numerical range of operators: convexity and related results, Studia Math, 194, 91-104, (2009) · Zbl 1178.47001 [3] Li, CK; Poon, YT, Generalized numerical ranges and quantum error correction, J Operator Theory, 66, 335-351, (2011) · Zbl 1261.47012 [4] Li, CK; Poon, YT, Convexity of the joint numerical range, SIAM J Matrix Anal Appl, 21, 668-678, (1999) [5] Arora, SC; Kumar, R, Joint essential numerical range, Glas Mat Ser III, 18, 317-320, (1983) · Zbl 0559.47003 [6] Kumar, R, A note on the boundary of the joint numerical range, Tamkang J Math, 14, 1-4, (1983) · Zbl 0518.47003 [7] Yang, Y, On extreme points of the joint numerical range, Tamkang J Math, 21, 399-403, (1990) · Zbl 0724.47003 [8] Garske, G, The boundary of the numerical range of an operator, J Math Anal Appl, 68, 605-607, (1979) · Zbl 0411.47001 [9] Takaguchi, M; Chō, M, The joint numerical range and the joint essential numerical range, Sci Rep Hirosaki Univ, 27, 6-8, (1980) · Zbl 0447.47005 [10] Martini, H; Wenzel, W, An analogue of the Krein-Milman theorem for star-shaped sets, Beitrage Algebra Geom, 44, 441-449, (2003) · Zbl 1043.52006 [11] Chō, M; Takaguchi, M, Boundary points of joint numerical ranges, Pac J Math, 95, 27-35, (1981) · Zbl 0502.47002 [12] Binding, P; Li, CK, Joint ranges of Hermitian matrices and simultaneous diagonalization, Linear Algebra Appl, 151, 157-167, (1991) · Zbl 0724.15020 [13] Das, KC, On stationary values of Rayleigh quotient of an operator, J Math Anal Appl, 48, 527-533, (1974) · Zbl 0297.47002 [14] Berberian, SK; Orland, GH, On the closure of the numerical range of an operator, Proc Am Math Soc, 18, 499-503, (1967) · Zbl 0173.42104 [15] Berberian, SK, Approximate proper vectors, Proc Am Math Soc, 13, 111-114, (1962) · Zbl 0166.40503 [16] Hansmann, M, An observation concerning boundary points of the numerical range, Oper Matrices, 9, 545-548, (2015) · Zbl 1329.47004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.