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1.
In this paper we prove that for any unital -weakly closed algebra which is -weakly generated by finite-rank operators in , every -weakly closed -submodule has . In the case of nest algebras, if are nests, we obtain the following -fold tensor product formula:


where each is the -weakly closed Alg -submodule determined by an order homomorphism from into itself.

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2.
Let be a -algebra acting on a Hilbert space , let be a linear mapping and let be a -derivation. Generalizing the celebrated theorem of Sakai, we prove that if is a continuous -mapping, then is automatically continuous. In addition, we show the converse is true in the sense that if is a continuous --derivation, then there exists a continuous linear mapping such that is a --derivation. The continuity of the so-called - -derivations is also discussed.

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3.

Given the disk algebra and an automorphism , there is associated a non-self-adjoint operator algebra called the semicrossed product of with . Buske and Peters showed that there is a one-to-one correspondence between the contractive Hilbert modules over and pairs of contractions and on satisfying . In this paper, we show that the orthogonally projective and Shilov Hilbert modules over correspond to pairs of isometries on satisfying . The problem of commutant lifting for is left open, but some related results are presented.  相似文献   


4.

We estimate double exponential sums of the form

where is of multiplicative order modulo the prime and and are arbitrary subsets of the residue ring modulo . In the special case , our bound is nontrivial for with any fixed 0$">, while if in addition we have it is nontrivial for .

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5.
A unital -algebra is said to have the (APD)-property if every nonzero element in has the approximate polar decomposition. Let be a closed ideal of . Suppose that and have (APD). In this paper, we give a necessary and sufficient condition that makes have (APD). Furthermore, we show that if and or is a simple purely infinite -algebra, then has (APD).

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6.
We identify iterated function systems and regular Borel measures such that the matrix subdivision process relative to a finite family converges if and only if satisfies certain spectral properties.  相似文献   

7.

We show that if is an -regular set in for which the triple integral of the Menger curvature is finite and if , then almost all of can be covered with countably many curves. We give an example to show that this is false for .

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8.
Let be bounded operators on a Hilbert space such that . Given a symmetry on , i.e., , we define the -symmetric commutant of to be the operator space

In this paper we obtain lifting theorems for symmetric commutants. The result extends the Sz.-Nagy-Foias commutant lifting theorem (), the anticommutant lifting theorem of Sebestyén ( ), and the noncommutative commutant lifting theorem ( ). Sarason's interpolation theorem for is extended to symmetric commutants on Fock spaces.  相似文献   


9.
In this paper, we first introduce the concept of single elements in a module. A systematic study of single elements in the Alg-module is initiated, where is a completely distributive subspace lattice on a Hilbert space . Furthermore, as an application of single elements, we study module isomorphisms between norm closed Alg-modules, where is a nest, and obtain the following result: Suppose that are norm closed Alg-modules and that is a module isomorphism. Then and there exists a non-zero complex number such that .

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10.
A -semigroup on a Banach space is called hypercyclic if there exists an element such that is dense in . is called chaotic if is hypercyclic and the set of its periodic vectors is dense in as well. We show that a spectral condition introduced by Desch, Schappacher and Webb requiring many eigenvectors of the generator which depend analytically on the eigenvalues not only implies the chaoticity of the semigroup but the chaoticity of every . Furthermore, we show that semigroups whose generators have compact resolvent are never chaotic. In a second part we prove the existence of hypercyclic vectors in for a hypercyclic semigroup , where is its generator.

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