For any n×n matrix A , we use the joint higher rank numerical range, Λk(A, . . . ,Am) , to define the higher rank numerical hull of A . We characterize the higher rank numerical hulls of Hermitian matrices. Also, the higher rank numerical hulls of unitary matrices are studied. Mathematics subject classification (2010): 15A60,81P68.

It is shown that for $n \le 3$ the joint numerical range of a family commuting $n\times n$ complex matrices always convex; \ge 4$ there are two whose not convex.

We consider linearly independent families of Hermitian matrices {A1, . . . , Am} so thatWk(A) is convex. It is shown that m can reach the upper bound 2k(n− k) + 1. A key idea in our study is relating the convexity of Wk(A) to the problem of constructing rank k orthogonal projections under linear constraints determined by A. The techniques are extended to study the convexity of other generalized...

let ‎x be an ‎‎n-‎‎‎‎‎‎square complex matrix with the ‎cartesian decomposition ‎‎x = a + i ‎b‎‎‎‎‎, ‎where ‎‎a ‎and ‎‎b ‎are ‎‎‎n ‎‎times n‎ ‎hermitian ‎matrices. ‎it ‎is ‎known ‎that ‎‎$vert x vert_p^2 ‎leq 2(vert a vert_p^2 + vert b vert_p^2)‎‎‎$, ‎where ‎‎$‎p ‎‎geq 2‎$‎ ‎and ‎‎$‎‎vert . vert_p$ ‎is ‎the ‎schatten ‎‎‎‎p-norm.‎ ‎‎ ‎‎in this paper‎, this inequality and some of its improvements ...