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1.
For any n-by-n matrix A  , we consider the maximum number k=k(A)k=k(A) for which there is a k-by-k compression of A   with all its diagonal entries in the boundary ∂W(A)W(A) of the numerical range W(A)W(A) of A. If A   is a normal or a quadratic matrix, then the exact value of k(A)k(A) can be computed. For a matrix A   of the form B⊕CBC, we show that k(A)=2k(A)=2 if and only if the numerical range of one summand, say, B is contained in the interior of the numerical range of the other summand C   and k(C)=2k(C)=2. For an irreducible matrix A  , we can determine exactly when the value of k(A)k(A) equals the size of A  . These are then applied to determine k(A)k(A) for a reducible matrix A   of size 4 in terms of the shape of W(A)W(A).  相似文献   

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This paper is devoted to a problem of finding the smallest positive integer s(m,n,k)s(m,n,k) such that (m+1)(m+1) generic skew-symmetric (k+1)(k+1)-forms in (n+1)(n+1) variables as linear combinations of the same s(m,n,k)s(m,n,k) decomposable skew-symmetric (k+1)(k+1)-forms.  相似文献   

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Let K   be a hypergroup with a Haar measure. The purpose of the present paper is to initiate a systematic approach to the study of the class of invariant complemented subspaces of L(K)L(K) and C0(K)C0(K), the class of left translation invariant w?w?-subalgebras of L(K)L(K) and finally the class of non-zero left translation invariant C?C?-subalgebras of C0(K)C0(K) in the hypergroup context with the goal of finding some relations between these function spaces. Among other results, we construct two correspondences: one, between closed Weil subhypergroups and certain left translation invariant w?w?-subalgebras of L(K)L(K), and another, between compact subhypergroups and a specific subclass of the class of left translation invariant C?C?-subalgebras of C0(K)C0(K). By the help of these two characterizations, we extract some results about invariant complemented subspaces of L(K)L(K) and C0(K)C0(K).  相似文献   

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Let (W,S)(W,S) be a Coxeter system with a strictly complete Coxeter graph. The present paper concerns the set Red(z)Red(z) of all reduced expressions for any z∈WzW. By associating each bc-expression to a certain symbol, we describe the set Red(z)Red(z) and compute its cardinal |Red(z)||Red(z)| in terms of symbols. An explicit formula for |Red(z)||Red(z)| is deduced, where the Fibonacci numbers play a crucial role.  相似文献   

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Let G   denote a locally compact Hausdorff group and M(G)M(G) be the space of all bounded complex-valued regular Borel measures on G  . In this paper, we define two strict topologies on M(G)M(G) and study various properties of these topologies such as metrizability, barrelledness and completeness. We also determine the dual space of M(G)M(G) and consider various continuity properties for the convolution product on M(G)M(G) under these topologies.  相似文献   

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