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1.
We study the relative position of several subspaces in a separable infinite-dimensional Hilbert space. In finite-dimensional case, Gelfand and Ponomarev gave a complete classification of indecomposable systems of four subspaces. We construct exotic examples of indecomposable systems of four subspaces in infinite-dimensional Hilbert spaces. We extend their Coxeter functors and defect using Fredholm index. The relative position of subspaces has close connections with strongly irreducible operators and transitive lattices. There exists a relation between the defect and the Jones index in a type II1 factor setting.  相似文献   

2.
We prove that n pairwise commuting derivations of the polynomial ring (or the power series ring) in n variables over a field k of characteristic 0 form a commutative basis of derivations if and only if they are k-linearly independent and have no common Darboux polynomials. This result generalizes a recent result due to Petravchuk and is an analogue of a well-known fact that a set of pairwise commuting linear operators on a finite dimensional vector space over an algebraically closed field has a common eigenvector.  相似文献   

3.
In the present note we give a new and short proof of Naimark's theorem asserting that for every commuting family ? of unitary operators in a πk-space Πk there exists ak-dimensional, nonpositive subspace invariant under ?.  相似文献   

4.
5.
We study the k-summability of divergent formal solutions for the Cauchy problem of a certain class of linear partial differential operators with time dependent coefficients. The problem is reduced to a k-summability property of formal solutions for a linear similar ordinary differential equation associated with the Cauchy problem.  相似文献   

6.
Pairs (A,B) of mutually annihilating operators AB=BA=0 on a finite dimensional vector space over an algebraically closed field were classified by Gelfand and Ponomarev [Russian Math. Surveys 23 (1968) 1-58] by method of linear relations. The classification of (A,B) over any field was derived by Nazarova, Roiter, Sergeichuk, and Bondarenko [J. Soviet Math. 3 (1975) 636-654] from the classification of finitely generated modules over a dyad of two local Dedekind rings. We give canonical matrices of (A,B) over any field in an explicit form and our proof is constructive: the matrices of (A,B) are sequentially reduced to their canonical form by similarity transformations (A,B)?(S-1AS,S-1BS).  相似文献   

7.
We consider commuting differential operators with two independent variables of general form and obtain general necessary commutativity conditions for low-order operators. We show that these conditions allow classifying commuting pairs of operators whose coefficients are linear functions of the independent variables.  相似文献   

8.
We present some result of lifting of the Gelfand Phillips property from Banach spacesE andF to Banach spaces of compact operators and of Bochner integrable functions. Moreover we studyC(K) spaces possessing the same property. In the last section we prove some result concerning the so called three space problem for the Gelfand Phillips property too.  相似文献   

9.
The kernel function of Cauchy type for type BC is defined as a solution of linear q-difference equations. In this paper, we show that this kernel function intertwines the commuting family of van Diejen’s q-difference operators. This result gives rise to a transformation formula for certain multiple basic hypergeometric series of type BC. We also construct a new infinite family of commuting q-difference operators for which the Koornwinder polynomials are joint eigenfunctions.  相似文献   

10.
It is well known that each pair of commuting linear operators on a finite dimensional vector space over an algebraically closed field has a common eigenvector. We prove an analogous statement for derivations of k[x] and k[x,y] over any field k of zero characteristic. In particular, if D1 and D2 are commuting derivations of k[x,y] and they are linearly independent over k, then either (i) they have a common polynomial eigenfunction; i.e., a nonconstant polynomial fk[x,y] such that D1(f)=λf and D2(f)=μf for some λ,μk[x,y], or (ii) they are Jacobian derivations
  相似文献   

11.
We consider the initial-boundary value problem (IBVP) for the Korteweg–de Vries equation with zero boundary conditions at x=0 and arbitrary smooth decreasing initial data. We prove that the solution of this IBVP can be found by solving two linear inverse scattering problems (SPs) on two different spectral planes. The first SP is associated with the KdV equation. The second SP is self-conjugate and its scattering function is found in terms of entries of the scattering matrix s(k) for the first SP. Knowing the scattering function, we solve the second inverse SP for finding the potential self-conjugate matrix. Consequently, the unknown object entering coefficients in the system of evolution equations for s(k,t) is found. Then, the time-dependent scattering matrix s(k,t) is expressed in terms of s(k)=s(k,0) and of solutions of the self-conjugate SP. Knowing s(k,t), we find the solution of the IBVP in terms of the solution of the Gelfand–Levitan–Marchenko equation in the first inverse SP.  相似文献   

12.
The S-spectrum has been introduced for the definition of the S-functional calculus that includes both the quaternionic functional calculus and a calculus for n-tuples of nonnecessarily commuting operators. The notion of right spectrum for right linear quaternionic operators has been widely used in the literature, especially in the context of quaternionic quantum mechanics. Moreover, several results in linear algebra, like the spectral theorem for quaternionic matrices, involve the right spectrum. In this Note we prove that the two notions of S-spectrum and of right spectrum coincide.  相似文献   

13.
We give the necessary and sufficient condition for a bounded linear operator with property (ω) by means of the induced spectrum of topological uniform descent, and investigate the permanence of property (ω) under some commuting perturbations by power finite rank operators. In addition, the theory is exemplified in the case of algebraically paranormal operators.  相似文献   

14.
The generalized eigenvalue problem for an arbitrary self-adjoint operator is solved in a Gelfand triple consisting of three Hilbert spaces. The proof is based on a measure theoretical version of the Sobolev lemma, and the multiplicity theory for self-adjoint operators. As an application necessary and sufficient conditions are mentioned such that a self-adjoint operator in L2(R) has (generalized) eigenfunctions which are tempered distributions.  相似文献   

15.
The operator norm of the derivative of the map which takes a finite-dimensional linear operator to its kth Grassman power (the kth compound) is evaluated. This leads to a bound for the distance between the Grassman powers of two operators. As an important application, a bound for the distance between the eigenvalues of two operators is obtained.  相似文献   

16.
We investigate n-tuples of commuting Foias-Williams/Peller type operators acting on vector-valued weighted Bergman spaces. We prove that a commuting n-tuple of such operators is jointly (completely) polynomially bounded if and only if it is similar to an n-tuple of contractions, if and only if each of the n operators is polynomially bounded.  相似文献   

17.
This paper considers the k-hyperexpansive Hilbert space operators T (those satisfying , 1?n?k) and the k-expansive operators (those satisfying the above inequality merely for n=k). It is known that if T is k-hyperexpansive then so is any power of T; we prove the analogous result for T assumed merely k-expansive. Turning to weighted shift operators, we give a characterization of k-expansive weighted shifts, and produce examples showing the k-expansive classes are distinct. For a weighted shift W that is k-expansive for all k (that is, completely hyperexpansive) we obtain results for k-hyperexpansivity of back step extensions of W. In addition, we discuss the completely hyperexpansive completion problem which is parallel to Stampfli's subnormal completion problem.  相似文献   

18.
We examine k-minimal and k-maximal operator spaces and operator systems, and investigate their relationships with the separability problem in quantum information theory. We show that the matrix norms that define the k-minimal operator spaces are equal to a family of norms that have been studied independently as a tool for detecting k-positive linear maps and bound entanglement. Similarly, we investigate the k-super minimal and k-super maximal operator systems that were recently introduced and show that their cones of positive elements are exactly the cones of k-block positive operators and (unnormalized) states with Schmidt number no greater than k, respectively. We characterize a class of norms on the k-super minimal operator systems and show that the completely bounded versions of these norms provide a criterion for testing the Schmidt number of a quantum state that generalizes the recently-developed separability criterion based on trace-contractive maps.  相似文献   

19.
We study vertex cut and flow sparsifiers that were recently introduced by Moitra (2009), and Leighton and Moitra (2010). We improve and generalize their results. We give a new polynomial-time algorithm for constructing O(log k/ log log k) cut and flow sparsifiers, matching the best known existential upper bound on the quality of a sparsifier, and improving the previous algorithmic upper bound of O(log2k/ log log k). We show that flow sparsifiers can be obtained from linear operators approximating minimum metric extensions. We introduce the notion of (linear) metric extension operators, prove that they exist, and give an exact polynomialtime algorithm for finding optimal operators.  相似文献   

20.
The joint spectral theory of a system of pairwise commuting self-adjoint left-invariant differential operators L1,…,Ln on a connected Lie group G is studied, under the hypothesis that the algebra generated by them contains a “weighted subcoercive operator” of ter Elst and Robinson (1998) [52]. The joint spectrum of L1,…,Ln in every unitary representation of G is characterized as the set of the eigenvalues corresponding to a particular class of (generalized) joint eigenfunctions of positive type of L1,…,Ln. Connections with the theory of Gelfand pairs are established in the case L1,…,Ln generate the algebra of K-invariant left-invariant differential operators on G for some compact subgroup K of Aut(G).  相似文献   

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