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1.
Ilwoo Cho 《Complex Analysis and Operator Theory》2008,2(1):1-28
The main purpose of this paper is to introduce several measures determined by a given finite directed graph. To construct
σ-algebras for those measures, we consider several algebraic structures induced by G; (i) the free semigroupoid of the shadowed graph (ii) the graph groupoid of G, (iii) the disgram set and (iv) the reduced diagram set . The graph measures determined by (i) is the energy measure measuing how much energy we spent when we have some movements on G. The graph measures determined by (iii) is the diagram measure measuring how long we moved consequently from the starting positions (which are
vertices) of some movements on G. The graph measures and determined by (ii) and (iv) are the (graph) groupoid measure and the (quotient-)groupoid measure, respectively. We show that
above graph measurings are invariants on shadowed graphs of finite directed graphs. Also, we will consider the reduced diagram
measure theory on graphs. In the final chapter, we will show that if two finite directed graphs G
1 and G
2 are graph-isomorphic, then the von Neumann algebras L
∞(μ
1) and L
∞(μ
2) are *-isomorphic, where μ
1 and μ
2 are the same kind of our graph measures of G
1 and G
2, respectively.
Received: December 7, 2006. Revised: August 3, 2007. Accepted: August 18, 2007. 相似文献
2.
Sebastian Bogner Bernd Fritzsche Bernd Kirstein 《Complex Analysis and Operator Theory》2007,1(1):55-95
The main theme of this paper is to characterize distinguished subclasses of the matricial Schur class
in terms of Taylor coefficients. Starting point of our investigations is the observation that the Taylor coefficient sequences
of functions from
are exactly the infinite p × q Schur sequences. We draw our attention mainly to the subclass
of
which consists of all p × q Schur functions for which the corresponding Taylor coefficient sequences are nondegenerate p × q Schur sequences. Using an appropriate adaptation of the Schur–Potapov algorithm for functions belonging to
to infinite sequences of complex p × q matrices we obtain an one-to-one correspondence between infinite nondegenerate p × q Schur sequences and the set of all infinite sequences (Ej)j=0∞ of strictly contractive complex p × q matrices. Taking into account the construction of
this gives us an one-to-one correspondence between
and the set of all infinite sequences (Ej)j=0∞ of strictly contractive complex p × q matrices. Hereby, (Ej)j =0∞ is called the sequence of Schur–Potapov parameters (shortly SP-parameters) of f.
Communicated by Daniel Alpay.
Submitted: August 17, 2006; Accepted: September 13, 2006 相似文献
3.
We prove Tolokonnikov’s Lemma and the inner-outer factorization for the real Hardy space
, the space of bounded holomorphic (possibly operator-valued) functions on the unit disc all of whose matrix-entries (with
respect to fixed orthonormal bases) are functions having real Fourier coefficients, or equivalently, each matrix entry f satisfies
for all z ∈
.
Tolokonnikov’s Lemma for
means that if f is left-invertible, then f can be completed to an isomorphism; that is, there exists an F, invertible in
, such that F = [ f f
c
] for some f
c
in
. In control theory, Tolokonnikov’s Lemma implies that if a function has a right coprime factorization over
, then it has a doubly coprime factorization in
. We prove the lemma for the real disc algebra
as well. In particular,
and
are Hermite rings.
The work of the first author was supported by Magnus Ehrnrooth Foundation.
Received: December 5, 2006. Revised: February 4, 2007. 相似文献
4.
Ilwoo Cho 《Complex Analysis and Operator Theory》2007,1(3):367-398
We identify two noncommutative structures naturally associated with countable directed graphs. They are formulated in the
language of operators on Hilbert spaces. If G is a countable directed graphs with its vertex set V(G) and its edge set E(G), then we associate partial isometries to the edges in E(G) and projections to the vertices in V(G). We construct a corresponding von Neumann algebra
as a groupoid crossed product algebra
of an arbitrary fixed von Neumann algebra M and the graph groupoid
induced by G, via a graph-representation (or a groupoid action) α. Graph groupoids are well-determined (categorial) groupoids. The graph
groupoid
of G has its binary operation, called admissibility. This
has concrete local parts
, for all e ∈ E(G). We characterize
of
, induced by the local parts
of
, for all e ∈ E(G). We then characterize all amalgamated free blocks
of
. They are chracterized by well-known von Neumann algebras: the classical group crossed product algebras
, and certain subalgebras
(M) of operator-valued matricial algebra
. This shows that graph von Neumann algebras identify the key properties of graph groupoids.
Received: December 20, 2006. Revised: March 07, 2007. Accepted: March 13, 2007. 相似文献
5.
M. Amélia Bastos Claudio A. Fernandes Yuri I. Karlovich 《Complex Analysis and Operator Theory》2008,2(2):241-272
The C*-subalgebra of generated by all multiplication operators by slowly oscillating and piecewise continuous functions, by the Cauchy singular
integral operator and by the range of a unitary representation of an amenable group of diffeomorphisms with any nonempty set of common fixed points is studied. A symbol calculus for the C*-algebra and a Fredholm criterion for its elements are obtained. For the C*-algebra composed by all functional operators in , an invertibility criterion for its elements is also established. Both the C*-algebras and are investigated by using a generalization of the local-trajectory method for C*-algebras associated with C*-dynamical systems which is based on the notion of spectral measure.
Submitted: April 30, 2007. Accepted: November 5, 2007. 相似文献
6.
In this paper we offer a computational approach to the spectral function for a finite family of commuting operators, and give
applications. Motivated by questions in wavelets and in signal processing, we study a problem about spectral concentration
of integral translations of functions in the Hilbert space . Our approach applies more generally to families of n arbitrary commuting unitary operators in a complex Hilbert space , or equivalent the spectral theory of a unitary representation U of the rank-n lattice in . Starting with a non-zero vector , we look for relations among the vectors in the cyclic subspace in generated by ψ. Since these vectors involve infinite “linear combinations,” the problem arises of giving geometric characterizations of these non-trivial linear
relations. A special case of the problem arose initially in work of Kolmogorov under the name L
2-independence. This refers to infinite linear combinations of integral translates of a fixed function with l
2-coefficients. While we were motivated by the study of translation operators arising in wavelet and frame theory, we stress
that our present results are general; our theorems are about spectral densities for general unitary operators, and for stochastic
integrals.
Work supported in part by the U.S. National Science Foundation. 相似文献
7.
Peter Ullrich 《Mathematische Zeitschrift》2007,255(4):827-846
We show that every tempered distribution, which is a solution of the (homogenous) Klein–Gordon equation, admits a “tame” restriction
to the characteristic (hyper)surface {x
0 + x
n
= 0} in (1 + n)-dimensional Minkowski space and is uniquely determined by this restriction. The restriction belongs to the space which we have introduced in (Ullrich in J. Math. Phys. 45, 2004). Moreover, we show that every element of appears as the “tame” restriction of a solution of the (homogeneous) Klein–Gordon equation. 相似文献
8.
Vasily A. Prokhorov Edward B. Saff Maxim Yattselev 《Complex Analysis and Operator Theory》2009,3(2):501-524
Let be a bounded simply connected domain with boundary Γ and let be a regular compact set with connected complement. In this paper we investigate asymptotics of the extremal constants:
where is the supremum norm on a compact set K, is the set of all algebraic polynomials of degree at most m, and as . Subsequently, we obtain asymptotic behavior of the Kolmogorov k-widths, , of the unit ball An∞ of restricted to E in C(E), where H∞ is the Hardy space of bounded analytic functions on G and C(E) is the space of continuous functions on E.
Received: April 24, 2008. Accepted: May 15, 2008. 相似文献
9.
Nikolai Tarkhanov 《Complex Analysis and Operator Theory》2007,1(1):115-141
We consider a boundary value problem for an elliptic differential operator of order 2m in a domain
. The boundary of
is smooth outside a smooth manifold Y of dimension 0 ≤ q < n − 1, and
bears edge type singularities along Y . The Lopatinskii condition is assumed to be fulfilled on the smooth part of
. The corresponding spaces are weighted Sobolev spaces
, and this allows one to define ellipticity of weight γ for the problem. The resolvent of the problem is assumed to possess
rays of minimal growth. The main result says that if there are rays of minimal growth with angles between neighbouring rays
not exceeding π(γ + 2m)/n, then the root functions of the problem are complete in
. In the case of second order elliptic equations the results remain true for all domains with Lipschitz boundary.
Communicated by Michael Shapiro.
Submitted: May 24, 2006; Accepted: June 15, 2006 相似文献
10.
Amol Sasane 《Complex Analysis and Operator Theory》2009,3(1):323-330
Let E be a separable infinite-dimensional Hilbert space, and let denote the algebra of all functions that are holomorphic. If is a subalgebra of , then using an algebraic result of Corach and Larotonda, we derive that under some conditions, the Bass stable rank of is infinite. In particular, we deduce that the Bass (and hence topological stable ranks) of the Hardy algebra , the disk algebra and the Wiener algebra are all infinite.
Submitted: October 10, 2007., Revised: January 11, 2008., Accepted: January 12, 2007. 相似文献
11.
Yury M. Arlinskiĭ Seppo Hassi Henk S. V. de Snoo 《Complex Analysis and Operator Theory》2007,1(2):211-233
Passive linear systems τ =
have their transfer function
in the Schur class S
. Using a parametrization of contractive block operators the transfer function
is connected to the Sz.-Nagy–Foiaş characteristic function
of the contraction A. This gives a new aspect and some explicit formulas for studying the interplay between the system τ and the functions
and
. The method leads to some new results for linear passive discrete-time systems. Also new proofs for some known facts in
the theory of these systems are obtained.
Dedicated to Eduard Tsekanovskiĭ on the occasion of his seventieth birthday
This work was supported by the Research Institute for Technology at the University of Vaasa.
The first author was also supported by the Academy of Finland (projects 212146, 117617) and the Dutch Organization for Scientific
Research N.W.O. (B 61-553).
Received: December 22, 2006. Revised: February 6, 2007. 相似文献
12.
Ameer Athavale 《Complex Analysis and Operator Theory》2008,2(3):417-428
Let be a strictly pseudoconvex bounded domain in with C
2 boundary . If a subnormal m-tuple T of Hilbert space operators has the spectral measure of its minimal normal extension N supported on , then T is referred to as a -isometry. Using some non-trivial approximation theorems in the theory of several complex variables, we establish a commutant
lifting theorem for those -isometries whose (joint) Taylor spectra are contained in a special superdomain Ω of . Further, we provide a function-theoretic characterization of those subnormal tuples whose Taylor spectra are contained in
Ω and that are quasisimilar to a certain (fixed) -isometry T (of which the multiplication tuple on the Hardy space of the unit ball in is a rather special example).
Submitted: September 9, 2007. Revised: October 10, 2007. Accepted: October 24, 2007. 相似文献
13.
Juha Lehrbäck 《manuscripta mathematica》2008,127(2):249-273
We establish necessary and sufficient conditions for a domain to admit the (p, β)-Hardy inequality , where d(x) = dist(x, ∂Ω) and . Our necessary conditions show that a certain dichotomy holds, even locally, for the dimension of the complement Ω
c
when Ω admits a Hardy inequality, whereas our sufficient conditions can be applied in numerous situations where at least
a part of the boundary ∂Ω is “thin”, contrary to previously known conditions where ∂Ω or Ω
c
was always assumed to be “thick” in a uniform way. There is also a nice interplay between these different conditions that
we try to point out by giving various examples.
The author was supported in part by the Academy of Finland. 相似文献
14.
Pascale Vitse 《Archiv der Mathematik》2005,85(4):374-385
For Banach space operators T satisfying the Tadmor-Ritt condition
a band limited H∞ calculus is established,
where
and a is at most of the order C(T)5. It follows that such a T allows a bounded Besov algebra B∞ 10 functional calculus,
These estimates are sharp in a convenient sense. Relevant embedding theorems for B∞ 10 are derived.
Received: 25 October 2004; revised: 31 January 2005 相似文献
15.
Yury M. Arlinskiĭ Seppo Hassi Henk S. V. de Snoo 《Complex Analysis and Operator Theory》2009,3(1):19-56
Passive systems with and as an input and output space and as a state space are considered in the case that the main operator on the state space is normal. Basic properties are given
and a general unitary similarity result involving some spectral theoretic conditions on the main operator is established.
A passive system with is said to be quasi-selfadjoint if ran . The subclass of the Schur class is the class formed by all transfer functions of quasi-selfadjoint passive systems. The subclass is characterized and minimal passive quasi-selfadjoint realizations are studied. The connection between the transfer function
belonging to the subclass and the Q-function of T is given.
Received: December 16, 2007., Accepted: March 4, 2008. 相似文献
16.
A. Rogozhin 《Integral Equations and Operator Theory》2007,57(2):283-301
In this paper we estimate the norm of the Moore-Penrose inverse T(a)+ of a Fredholm Toeplitz operator T(a) with a matrix-valued symbol a∈LN × N∞ defined on the complex unit circle. In particular, we show that in the ”generic case” the strict inequality ||T(a)+|| > ||a−1||∞ holds. Moreover, we discuss the asymptotic behavior of ||T(tra)+|| for
. The results are illustrated by numerical experiments. 相似文献
17.
J. J. Grobler 《Integral Equations and Operator Theory》2007,57(1):83-99
For a probability space
we denote the marginal measures of
, defined on Σ and Λ respectively, by
and
. If ρ is a function norm defined on
marginal function norms ρ1 and ρ2 are defined on
and
. We find conditions which guarantee Lρ 1 + Lρ 2 to be embedded in Lρ as a closed subspace. The problem is encountered in Statistics when estimating a bivariate distribution with known marginals.
We find a condition which, applied to the binormal distribution in L2, improves some known conditions. 相似文献
18.
Hari Bercovici 《Complex Analysis and Operator Theory》2007,1(3):335-339
Consider a domain
, and two analytic matrix-valued functions functions
. Consider also points
and positive integers n
1, n
2, . . . , n
N
. We are interested in the existence of an analytic function
such that X(ω) is invertible, and G(ω) coincides with X(ω)F(ω)X(ω)−1 up to order n
j
at the point ω
j
. We will see that such a function exists provided that F(ω
j
),G(ω
j
) have cyclic vectors, and the characteristic polynomials of F,G coincide up to order n
j
at ω
j
. This allows one to give a short proof to a result of Huang, Marcantognini and Young concerning spectral interpolation in
the unit disk.
The author was partially supported by a grant from the National Science Foundation.
Received: September 8, 2006. Accepted: January 11, 2007. 相似文献
19.
J. Sunklodas 《Lithuanian Mathematical Journal》2009,49(2):216-221
In the paper, we present upper bounds of L
p
norms of order (
X)-1/2 for all 1 ≤ p ≤ ∞ in the central limit theorem for a standardized random variable (X−
X)/ √
X, where a random variable X is distributed by the Poisson distribution with parameter λ > 0 or by the standard gamma distribution Γ(α, 0, 1) with parameter
α > 0.
The research was partially supported by the Lithuanian State Science and Studies Foundation, grant No. T-70/09. 相似文献
20.
Some Properties of Essential Spectra of a Positive Operator 总被引:1,自引:1,他引:0
Egor A. Alekhno 《Positivity》2007,11(3):375-386
Let E be a Banach lattice, T be a bounded operator on E. The Weyl essential spectrum σew(T) of the operator T is a set
, where
is a set of all compact operators on E. In particular for a positive operator T next subsets of the spectrum
are introduced in the article. The conditions by which
implies either
or
are investigated, where σef(T) is the Fredholm essential spectrum. By this reason, the relations between coefficients of the main part of the Laurent series
of the resolvent R(., T) of a positive operator T around of the point λ = r(T) are studied. The example of a positive integral operator T : L1→ L∞ which doesn’t dominate a non-zero compact operator, is adduced. Applications of results which are obtained, to the spectral
theory of band irreducible operators, are given. Namely, the criteria when the operator inequalities 0 ≤ S < T imply the spectral radius inequality r(S) < r(T), are established, where T is a band irreducible abstract integral operator. 相似文献