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1.
Bebe Prunaru 《Proceedings of the American Mathematical Society》1997,125(6):1689-1691
We show that if is a bounded operator on a Hilbert space such that for every polynomial , then has a nontrivial invariant subspace.
2.
There is a subtle difference as far as the invariant subspace problem is concerned for operators acting on real Banach spaces
and operators acting on complex Banach spaces. For instance, the classical hyperinvariant subspace theorem of Lomonosov [Funktsional. Anal. nal. i Prilozhen 7(3)(1973), 55–56. (Russian)], while true for complex Banach spaces is false for real Banach spaces. When one starts with a
bounded operator on a real Banach space and then considers some “complexification technique” to extend the operator to a complex
Banach space, there seems to be no pattern that indicates any connection between the invariant subspaces of the “real” operator
and those of its “complexifications.” The purpose of this note is to examine two complexification methods of an operator T acting on a real Banach space and present some questions regarding the invariant subspaces of T and those of its complexifications
Mathematics Subject Classification 1991: 47A15, 47C05, 47L20, 46B99
Y.A. Abramovich: 1945–2003
The research of Aliprantis is supported by the NSF Grants EIA-0075506, SES-0128039 and DMI-0122214 and the DOD Grant ACI-0325846 相似文献
3.
The paper studies the existence of closed invariant subspaces for a Lie algebra L of bounded operators on an infinite-dimensional Banach space X. It is assumed that L contains a Lie subalgebra L0 that has a non-trivial closed invariant subspace in X of finite codimension or dimension. It is proved that L itself has a non-trivial closed invariant subspace in the following two cases: (1) L0 has finite codimension in L and there are Lie subalgebras L0=L0⊂L1⊂?⊂Lp=L such that Li+1=Li+[Li,Li+1] for all i; (2) L0 is a Lie ideal of L and dim(L0)=∞. These results are applied to the problem of the existence of non-trivial closed Lie ideals and closed characteristic Lie ideals in an infinite-dimensional Banach Lie algebra L that contains a non-trivial closed Lie subalgebra of finite codimension. 相似文献
4.
William T. Ross 《Proceedings of the American Mathematical Society》1996,124(6):1841-1846
In this paper, we comment on the complexity of the invariant subspaces (under the bilateral Dirichlet shift ) of the harmonic Dirichlet space . Using the sampling theory of Seip and some work on invariant subspaces of Bergman spaces, we will give examples of invariant subspaces with , . We will also generalize this to the Dirichlet classes , , as well as the Besov classes , , .
5.
Sobolev圆盘代数的不变子空间 总被引:1,自引:1,他引:0
研究了Sobolev圆盘代数R(D)上乘自变量算子M_z的不变子空间,给出了M_z在任何不变子空间上限制的基本性质,证明了M_z分别限制在两个不变子空间上酉等价当且仅当这两个不变子空间相等,并描述了M_z的一类公共零点在边界的不变子空间的结构. 相似文献
6.
研究了一类二阶二次变系数微分算子的不变子空间,讨论了这类微分算子不变子空间的应用,并给出了具体应用的一些例子.在这些例子中,构造了大量变系数非线性演化方程的精确解. 相似文献
7.
Mathematical Notes - 相似文献
8.
Gabriel H. Tucci 《Journal of Functional Analysis》2008,254(12):2969-2994
For each sequence n{cn} in l1(N) we define an operator A in the hyperfinite II1-factor R. We prove that these operators are quasinilpotent and they generate the whole hyperfinite II1-factor. We show that they have non-trivial, closed, invariant subspaces affiliated to the von Neumann algebra and we provide enough evidence to suggest that these operators are interesting for the hyperinvariant subspace problem. We also present some of their properties. In particular, we show that the real and imaginary part of A are equally distributed, and we find a combinatorial formula as well as an analytical way to compute their moments. We present a combinatorial way of computing the moments of A∗A. 相似文献
9.
10.
Let H2(D2) be the Hardy space over the bidisk. For sequences of Blaschke products {φn(z):−∞<n<∞} and {ψn(w):−∞<n<∞} satisfying some additional conditions, we may define a Rudin type invariant subspace M. We shall determine the rank of H2(D2)?M for the pair of operators and . 相似文献
11.
It is proved that if a Lie algebra of compact operators contains a nonzero ideal consisting of quasinilpotent operators then this Lie algebra has a nontrivial invariant subspace. Some applications of this result to lattices of invariant subspaces for families of compact operators and to structures of ideals of Banach Lie algebras with compact adjoint action are given. 相似文献
12.
Armando Machado 《Linear and Multilinear Algebra》2013,61(6):561-566
We present an elementary proof of the existence of an eigenvalue for an endomorphism of a complex vector space and we derive the Fundamental Theorem of Algebra as a corollary of this existence. We also present new proofs for the corresponding results for endomorphisms of real vector spaces. 相似文献
13.
Alexandru Aleman Stefan Richter Carl Sundberg 《Transactions of the American Mathematical Society》2007,359(7):3369-3407
Let be a Hilbert space of analytic functions on the open unit disc such that the operator of multiplication with the identity function defines a contraction operator. In terms of the reproducing kernel for we will characterize the largest set such that for each , the meromorphic function has nontangential limits a.e. on . We will see that the question of whether or not has linear Lebesgue measure 0 is related to questions concerning the invariant subspace structure of .
We further associate with a second set , which is defined in terms of the norm on . For example, has the property that for all if and only if has linear Lebesgue measure 0.
It turns out that a.e., by which we mean that has linear Lebesgue measure 0. We will study conditions that imply that a.e.. As one corollary to our results we will show that if dim and if there is a such that for all and all we have , then a.e. and the following four conditions are equivalent:
(1) for some ,
(2) for all , ,
(3) has nonzero Lebesgue measure,
(4) every nonzero invariant subspace of has index 1, i.e., satisfies dim .
14.
Yu. O. Golovin 《Mathematical Notes》1998,63(1):9-18
This paper is concerned with the connection between the geometric properties of the latticeL of subspaces of a Hilbert spaceH and homological properties (flatness and injectivity) ofH regarded as a natural module over the reflexive algebra AlgL that consists of all operators leaving invariant each element of the latticeL. It follows from these results that the cohomology groups with coefficients inB(H) are trivial for a broad class of reflexive algebras.
Translated fromMatemalicheskie Zametki, Vol. 63, No. 1, pp. 9–20, January, 1998.
The author gladly expresses his sincere gratitude to A. Ya. Khelemskii and Yu. V. Selivanov for their assistance in this work.
In particular, A. Ya. Khelemskii indicated how to simplify the proof of Theorem 1.
This research was supported by the Russian Foundation for Basic Research under grant No. 93-01-00156, by the International
Science Foundation under grant No. M95000, and by the INTAS fund under grant No. 93-1376. 相似文献
15.
16.
A. A. Shkalikov 《Functional Analysis and Its Applications》2007,41(2):154-167
We prove that a dissipative operator in the Krein space has a maximal nonnegative invariant subspace provided that the operator admits matrix representation with respect to the canonical decomposition of the space and the upper right operator in this representation is compact relative to the lower right operator. Under the additional assumption that the upper and lower left operators are bounded (the so-called Langer condition), this result was proved (in increasing order of generality) by Pontryagin, Krein, Langer, and Azizov. We relax the Langer condition essentially and prove under the new assumptions that a maximal dissipative operator in the Krein space has a maximal nonnegative invariant subspace such that the spectrum of its restriction to this subspace lies in the left half-plane. Sufficient conditions are found for this restriction to be the generator of a holomorphic semigroup or a C 0-semigroup. 相似文献
17.
Brent J. Carswell 《Proceedings of the American Mathematical Society》2003,131(4):1233-1241
In both the Bergman space and the Hardy space , the problem of determining which bounded univalent mappings of the unit disk have the wandering property is addressed. Generally, a function in has the wandering property in , where denotes either or , provided that every -invariant subspace of is generated by the orthocomplement of within . It is known that essentially every function which has the wandering property in either space is the composition of a univalent mapping with a classical inner function, and that the identity mapping has this property in both spaces. Consequently, weak-star generators of also have the wandering property in both settings. The present paper gives a partial converse to this, and shows that in both settings there is a large class of bounded univalent mappings which fail to have the wandering property.
18.
Daniel Turcotte 《Proceedings of the American Mathematical Society》2003,131(5):1399-1404
Globevnik and Vidav have studied the propagation of normality from an open subset of a region of the complex plane for analytic functions with values in the space of bounded linear operators on a Hilbert space . We obtain a propagation of normality in the more general setting of a converging sequence located on a regular analytic Jordan arc in the complex plane for analytic functions with values in a complex unital Banach algebra with continuous involution. We show that in this more general setting, the propagation of normality does not imply functional commutativity anymore as it does in the case studied by Globevnik and Vidav. An immediate consequence of the Propagation of Normality Theorem is that the further generalization given by Wolf of Jamison's generalization of Rellich's theorem is equivalent to Jamison's result. We obtain a propagation property within Banach subspaces for analytic Banach space-valued functions. The propagation of normality differs from the propagation within Banach subspaces since the set of all normal elements does not form a Banach subspace.
19.
In this paper, we study the Banach algebra
generated by multidimensional pair integral operators with homogeneous kernels. We describe necessary and sufficient conditions for operators from the algebra
to be Fredholm and present a formula for calculating the index. 相似文献
20.
A. T. Gainov 《Algebra and Logic》2007,46(4):231-243
The concept of a composition algebra of the second kind is introduced. We prove that such algebras are non-degenerate monocomposition
algebras without unity. A big number of these algebras in any finite dimension are constructed, as well as two algebras in
a countable dimension. The constructed algebras each contains a non-isotropic idempotent e2 = e. We describe all orthogonally non-isomorphic composition algebras of the second kind in the following forms: (1) a two-dimensional
algebra (which has turned out to be unique); (2) three-dimensional algebras in the constructed series. For every algebra A,
the group Ortaut A of orthogonal automorphisms is specified.
__________
Translated from Algebra i Logika, Vol. 46, No. 4, pp. 428–447, July–August, 2007. 相似文献