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1.
Let G be a graph with n vertices and m edges and let μ(G) = μ1(G) ? ? ? μn(G) be the eigenvalues of its adjacency matrix. Set s(G)=∑uV(G)d(u)-2m/n∣. We prove that
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2.
Let Cp be the Schatten p-class for p>0. Generalizations of the parallelogram law for the Schatten 2-norms have been given in the following form: if A={A1,A2,…,An} and B={B1,B2,…,Bn} are two sets of operators in, then C2
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3.
Bergman-Privalov class ANα(B) consists of all holomorphic functions on the unit ball BCn such that
fANα:=∫Bln(1+∣f(z)∣)dVα(z)<,  相似文献   

4.
Let B(H) be the algebra of bounded linear operator acting on a Hilbert space H (over the complex or real field). Characterization is given to A1,…,AkB(H) such that for any unitary operators is always in a special class S of operators such as normal operators, self-adjoint operators, unitary operators. As corollaries, characterizations are given to AB(H) such that complex, real or nonnegative linear combinations of operators in its unitary orbit U(A)={UAU:Uunitary} always lie in S.  相似文献   

5.
Let T1,…,Td be linear contractions on a complex Hilbert space and p a complex polynomial in d variables which is a sum of d single variable polynomials. We show that the operator norm of p(T1,…,Td) is bounded by
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6.
A Hilbert space operator AB(H) is p-hyponormal, A∈(p-H), if |A|2p?|A|2p; an invertible operator AB(H) is log-hyponormal, A∈(?-H), if log(TT)?log(TT). Let dAB=δAB or ?AB, where δABB(B(H)) is the generalised derivation δAB(X)=AX-XB and ?ABB(B(H)) is the elementary operator ?AB(X)=AXB-X. It is proved that if A,B∈(?-H)∪(p-H), then, for all complex λ, , the ascent of (dAB-λ)?1, and dAB satisfies the range-kernel orthogonality inequality ‖X‖?‖X-(dAB-λ)Y‖ for all X∈(dAB-λ)-1(0) and YB(H). Furthermore, isolated points of σ(dAB) are simple poles of the resolvent of dAB. A version of the elementary operator E(X)=A1XA2-B1XB2 and perturbations of dAB by quasi-nilpotent operators are considered, and Weyl’s theorem is proved for dAB.  相似文献   

7.
For any operator A on a Hilbert space, let W(A), w(A) and w0(A) denote its numerical range, numerical radius and the distance from the origin to the boundary of its numerical range, respectively. We prove that if An=0, then w(A)?(n-1)w0(A), and, moreover, if A attains its numerical radius, then the following are equivalent: (1) w(A)=(n-1)w0(A), (2) A is unitarily equivalent to an operator of the form aAnA, where a is a scalar satisfying |a|=2w0(A), An is the n-by-n matrix
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8.
Let X be a real finite-dimensional normed space with unit sphere SX and let L(X) be the space of linear operators from X into itself. It is proved that X is an inner product space if and only if for A,CL(X)
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9.
Let A and B be bounded linear operators acting on a Hilbert space H. It is shown that the triangular inequality serves as the ultimate estimate of the upper norm bound for the sum of two operators in the sense that
sup{∥U*AU+V*BV∥:U and V are unitaries}=min{∥A+μI∥+∥B-μI∥:μC}.  相似文献   

10.
11.
Let A1, … , Ak be positive semidefinite matrices and B1, … , Bk arbitrary complex matrices of order n. We show that
span{(A1x)°(A2x)°?°(Akx)|xCn}=range(A1°A2°?°Ak)  相似文献   

12.
We show consistency and asymptotic normality of certain estimators for expected exponential growth rates under i.i.d. observations. These statistical functionals are of the form
T(F)=∫logh(x,y)F(dx)F(dy)  相似文献   

13.
Let (Ω,Σ,μ) a measure space such that 0<μ(A)<1<μ(B)<∞ for some A,BΣ. Under some natural conditions on the bijective functions φ,φ1,φ2,ψ,ψ1,ψ2:(0,∞)→(0,∞) we prove that if
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14.
We point out a sharp reverse Cauchy-Schwarz/Hölder matrix inequality. The Cauchy-Schwarz version involves the usual matrix geometric mean: Let Ai and Bi be positive definite matrices such that 0<mAi?Bi?MAi for some scalars 0<m?M and i=1,2,?,n. Then
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15.
Let 1 ? p ? ∞, 0 < q ? p, and A = (an,k)n,k?0 ? 0. Denote by Lp,q(A) the supremum of those L satisfying the following inequality:
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16.
We show that every injective Jordan semi-triple map on the algebra Mn(F) of all n × n matrices with entries in a field F (i.e. a map Φ:Mn(F)→Mn(F) satisfying
Φ(ABA)=Φ(A)Φ(B)Φ(A)  相似文献   

17.
Maps completely preserving spectral functions   总被引:1,自引:0,他引:1  
Let X,Y be infinite dimensional complex Banach spaces and A,B be standard operator algebras on X and Y, respectively. In this paper, we show that surjective maps completely preserving certain spectral function Δ(·) from A to B are isomorphisms, where Δ(·) stands for any one of 13 spectral functions σ(·), σl(·), σr(·), σl(·)∩σr(·), σ(·), ησ(·), σp(·), σc(·), σp(·)∩σc(·), σp(·)∪σc(·), σap(·), σs(·), and σap(·)∩σs(·).  相似文献   

18.
In this paper, we present a complement of a generalized Ando-Hiai inequality due to Fujii and Kamei [M. Fujii, E. Kamei, Ando-Hiai inequality and Furuta inequality, Linear Algebra Appl. 416 (2006) 541-545]. Let A and B be positive operators on a Hilbert space H such that 0<m1?A?M1 and 0<m2?B?M2 for some scalars mi?Mi (i=1,2), and let α∈[0,1]. Put for i=1,2. Then for each 0<r?1 and s?1
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19.
Let B(X) be the algebra of all bounded linear operators on the Banach space X, and let N(X) be the set of nilpotent operators in B(X). Suppose ?:B(X)→B(X) is a surjective map such that A,BB(X) satisfy ABN(X) if and only if ?(A)?(B)∈N(X). If X is infinite dimensional, then there exists a map f:B(X)→C?{0} such that one of the following holds:
(a)
There is a bijective bounded linear or conjugate-linear operator S:XX such that ? has the form A?S[f(A)A]S-1.
(b)
The space X is reflexive, and there exists a bijective bounded linear or conjugate-linear operator S : X′ → X such that ? has the form A ? S[f(A)A′]S−1.
If X has dimension n with 3 ? n < ∞, and B(X) is identified with the algebra Mn of n × n complex matrices, then there exist a map f:MnC?{0}, a field automorphism ξ:CC, and an invertible S ∈ Mn such that ? has one of the following forms:
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20.
We give tight lower bounds on the cardinality of the sumset of two finite, nonempty subsets A,BR2 in terms of the minimum number h1(A,B) of parallel lines covering each of A and B. We show that, if h1(A,B)?s and |A|?|B|?2s2−3s+2, then
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