2| | | A1/2XB1/2| | | £ | | | AvXB1-v+A1-vXBv| | | £ | | | AX+XB| | |.2\left\vert \left\vert \left\vert A^{1/2}XB^{1/2}\right\vert \right\vert \right\vert \leq \left\vert \left\vert \left\vert A^{v}XB^{1-v}+A^{1-v}XB^{v}\right\vert \right\vert \right\vert \leq \left\vert \left\vert \left\vert AX+XB\right\vert \right\vert \right\vert. 相似文献
5.
We show that the symmetrized product AB + BA of two positive operators A and B is positive if and only if f( A+ B) £ f( A)+ f( B){f(A+B)\leq f(A)+f(B)} for all non-negative operator monotone functions f on [0,∞) and deduce an operator inequality. We also give a necessary and sufficient condition for that the composition f ° g{f \circ g} of an operator convex function f on [0,∞) and a non-negative operator monotone function g on an interval ( a, b) is operator monotone and present some applications. 相似文献
6.
该文给出了四元数矩阵方程组X_1B_1=C_1,X_2B_2=C2,A_1X_1B_3+A_2X_2B_4=C_b可解的充要条件及其通解的表达式,利用此结果建立了四元数矩阵方程组XB_a=C_a,A_bXB_b=C_b有广义(反)反射解的充要条件及其有此种解时通解的表达式. 相似文献
7.
Universal C*-algebras C*(A) exist for certain topological *-algebras called algebras with a C*-enveloping algebra. A Frechet *-algebra A has a C*-enveloping algebra if and only if every operator representation of A maps A into bounded operators. This is proved by showing that every unbounded operator representation π, continuous in the uniform
topology, of a topological *-algebra A, which is an inverse limit of Banach *-algebras, is a direct sum of bounded operator representations, thereby factoring through
the enveloping pro- C*-algebra E(A) of A. Given a C*-dynamical system ( G,A,α), any topological *-algebra B containing C
c
( G,A) as a dense *-subalgebra and contained in the crossed product C*-algebra C*( G,A,α) satisfies E( B) = C*( G,A,α). If G = ℝ, if B is an α-invariant dense Frechet *-subalgebra of A such that E( B) = A, and if the action α on B is m-tempered, smooth and by continuous *-automorphisms: then the smooth Schwartz crossed product S(ℝ, B,α) satisfies E( S(ℝ, B,α)) = C*(ℝ, A,α). When G is a Lie group, the C
∞-elements C
∞( A), the analytic elements C
ω( A) as well as the entire analytic elements C
є( A) carry natural topologies making them algebras with a C*-enveloping algebra. Given a non-unital C*-algebra A, an inductive system of ideals I
α is constructed satisfying A = C*-ind lim I
α; and the locally convex inductive limit ind lim I
α is an m-convex algebra with the C*-enveloping algebra A and containing the Pedersen ideal K
a
of A. Given generators G with weakly Banach admissible relations R, we construct universal topological *-algebra A( G, R) and show that it has a C*-enveloping algebra if and only if ( G, R) is C*-admissible. 相似文献
8.
Summary Let X be a complex Hilbert space, let L( X) be the algebra of all bounded linear operators on X, and let A( X) ⊂ L( X) be a standard operator algebra, which is closed under the adjoint operation. Suppose there exists a linear mapping D: A( X) → L( X) satisfying the relation D( AA* A) = D( A) A* A + AD( A*) A + AA* D( A), for all A ∈ A( X). In this case D is of the form D( A) = AB- BA, for all A∈ A( X) and some B ∈ L( X), which means that D is a derivation. We apply this result to semisimple H*-algebras. 相似文献
9.
Let { R
n
}
n≥0 be a binary linear recurrence defined by R
n+2 = A
R
n+1 + B
R
n
( n ≥ 0), where A, B, R
0, R
1 are integers and Δ = A
2 + 4 B > 0. We give necessary and sufficient conditions for the transcendence of the numbers
?k 3 0¢\fracakRrk+b,\sum_{k\geq 0}{}^{\prime}\frac{a_k}{R_{r^k}+b}, 相似文献
10.
Equations such as AB = B
T
A have been studied in the finite dimensional setting in
(Linear Algebra Appl 369:279–294, 2003). These equations have implications for the spectrum of B, when A is normal. Our aim is to generalize these results to an infinite dimensional setting. In this case it is natural to use JB* J for some conjugation operator J in place of B
T
. Our main result is a spectral pairing theorem for a bounded normal operator B which is applied to the study of the equation KB = B* K for K an antiunitary operator. In particular, using conjugation operators, we generalize the notion of Hamiltonian operator and
skew-Hamiltonian operator in a natural way, derive some of their properties, and give a characterization of certain operators
B for which AB = ( JB* J) A and BA = A( JB* J) and also those B with KB = B* K for certain antiunitary operators K. 相似文献
11.
The consistent conditions and the general expressions about the Hermitian solutions of the linear matrix equations AXB= C and ( AX, XB)=( C, D) are studied in depth, where A, B, C and D are given matrices of suitable sizes. The Hermitian minimum F‐norm solutions are obtained for the matrix equations AXB= C and ( AX, XB)=( C, D) by Moore–Penrose generalized inverse, respectively. For both matrix equations, we design iterative methods according to the fundamental idea of the classical conjugate direction method for the standard system of linear equations. Numerical results show that these iterative methods are feasible and effective in actual computations of the solutions of the above‐mentioned two matrix equations. Copyright © 2006 John Wiley & Sons, Ltd. 相似文献
12.
It is well known that the Sylvester matrix equation AX + XB = C has a unique solution X if and only if 0 ∉ spec( A) + spec( B). The main result of the present article are explicit formulas for the determinant of X in the case that C is one-dimensional. For diagonal matrices A, B, we reobtain a classical result by Cauchy as a special case.The formulas we obtain are a cornerstone in the asymptotic classification of multiple pole solutions to integrable systems like the sine-Gordon equation and the Toda lattice. We will provide a concise introduction to the background from soliton theory, an operator theoretic approach originating from work of Marchenko and Carl, and discuss examples for the application of the main results. 相似文献
13.
This paper is concerned with perturbation problems of regularity linear systems. Two types of perturbation results are proved:
(i) the perturbed system ( A + P, B, C) generates a regular linear system provided both ( A, B, C) and ( A, B, P) generate regular linear systems; and (ii) the perturbed system (( A-1+D A)| X, B, CAL){((A_{-1}+\Delta A)|_X,B,C^A_\Lambda)} generates a regular linear system if both ( A, B, C) and ( A, Δ A, C) generate regular linear systems. These allow us to establish a new variation of constants formula of the control system
( A + P, B). Moreover, these results are applied to the linear systems with state and output delays. The regularity and the mild expressibility
is deduced, and a necessary and sufficient condition for stabilizability of the delayed systems is proved. 相似文献
14.
It is shown that if A, B, X are Hilbert space operators such that X? γI, for the positive real number γ, and p, q>1 with 1/ p+1/ q=1, then | A− B| 2? p| A| 2+ q| B| 2 with equality if and only if (1− p) A= B and γ|||| A− B| 2|||?||| p| A| 2X+ qX| B| 2||| for every unitarily invariant norm. Moreover, if in addition A, B are normal and X is any Hilbert-Schmidt operator, then ‖ δA,B2( X)‖ 2?‖ p| A| 2X+ qX| B| 2‖ 2 with equality if and only if (1− p) AX= XB. 相似文献
15.
In this paper we consider a nonlinear evolution reaction–diffusion system governed by multi-valued perturbations of m-dissipative operators, generators of nonlinear semigroups of contractions. Let X and Y be real Banach spaces, ${\mathcal{K}}In this paper we consider a nonlinear evolution reaction–diffusion system governed by multi-valued perturbations of m-dissipative operators, generators of nonlinear semigroups of contractions. Let X and Y be real Banach spaces, K{\mathcal{K}} be a nonempty and locally closed subset in
\mathbbR ×X×Y, A:D(A) í X\rightsquigarrow X, B:D(B) í Y\rightsquigarrow Y{\mathbb{R} \times X\times Y,\, A:D(A)\subseteq X\rightsquigarrow X, B:D(B)\subseteq Y\rightsquigarrow Y} two m-dissipative operators, F:K ? X{F:\mathcal{K} \rightarrow X} a continuous function and
G:K \rightsquigarrow Y{G:\mathcal{K} \rightsquigarrow Y} a nonempty, convex and closed valued, strongly-weakly upper semi-continuous (u.s.c.) multi-function. We prove a necessary
and a sufficient condition in order that for each (t,x,h) ? K{(\tau,\xi,\eta)\in \mathcal{K}}, the next system
{ lc u¢(t) ? Au(t)+F(t,u(t),v(t)) t 3 tv¢(t) ? Bv(t)+G(t,u(t),v(t)) t 3 tu(t)=x, v(t)=h, \left\{ \begin{array}{lc} u'(t)\in Au(t)+F(t,u(t),v(t))\quad t\geq\tau \\ v'(t)\in Bv(t)+G(t,u(t),v(t))\quad t\geq\tau \\ u(\tau)=\xi,\quad v(\tau)=\eta, \end{array} \right. 相似文献
16.
In this article we establish necessary and sufficient conditions for the existence and the expressions of the general real solutions to the classical system of quaternion matrix equations A 1 XB 1 = C 1, A 2 XB 2 = C 2. Moreover, formulas of the maximal and minimal ranks of four real matrices X 1, X 2, X 3, and X 4 in solution X = X 1 + X 2 i + X 3 j + X 4 k to the system mentioned above are derived. As applications, we give necessary and sufficient conditions for the quaternion matrix equations A 1 XB 1 = C 1, A 2 XB 2 = C 2, A 3 XB 3 = C 3 to have common real solutions. In addition, the maximal and minimal ranks of four real matrices E, F, G, and H in the common generalized inverse of A 1 + B 1 i + C 1 j + D 1 k and A 2 + B 2 i + C 2 j + D 2 k, which can be expressed as E + Fi + Gj + Hk are also presented. 相似文献
17.
The perturbation classes problem for semi-Fredholm operators asks when the equalities SS( X, Y)= PF +( X, Y){\mathcal{SS}(X,Y)=P\Phi_+(X,Y)} and SC( X, Y)= PF -( X, Y){\mathcal{SC}(X,Y)=P\Phi_-(X,Y)} are satisfied, where SS{\mathcal{SS}} and SC{\mathcal{SC}} denote the strictly singular and the strictly cosingular operators, and PΦ + and PΦ − denote the perturbation classes for upper semi-Fredholm and lower semi-Fredholm operators. We show that, when Y is a reflexive Banach space, SS( Y*, X*)= PF +( Y*, X*){\mathcal{SS}(Y^*,X^*)=P\Phi_+(Y^*,X^*)} if and only if SC( X, Y)= PF -( X, Y),{\mathcal{SC}(X,Y)=P\Phi_-(X,Y),} and SC( Y*, X*)= PF -( Y*, X*){\mathcal{SC}(Y^*,X^*)=P\Phi_-(Y^*,X^*)} if and only if SS( X, Y)= PF +( X, Y){\mathcal{SS}(X,Y)=P\Phi_+(X,Y)}. Moreover we give examples showing that both direct implications fail in general. 相似文献
19.
Let L( H) denote the algebra of all bounded linear operators on a separable infinite dimensional complex Hilbert space H into itself. Given A ∈ L( H), we define the elementary operator Δ
A
: L( H) → L( H) by Δ
A
( X) = AXA − X. In this paper we study the class of operators A ∈ L( H) which have the following property: ATA = T implies AT* A = T* for all trace class operators T ∈ C
1( H). Such operators are termed generalized quasi-adjoints. The main result is the equivalence between this character and the
fact that the ultraweak closure of the range of Δ
A
is closed under taking adjoints. We give a characterization and some basic results concerning generalized quasi-adjoints
operators. 相似文献
20.
The solution of the linear operator equation: An-1X+ An-2XB++ AXBn-2+ XBn-1= Y is given by if the spectra of A and B are in the sector { z: z≠0,- π/ n<arg z< π/ n}. 相似文献
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