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1.
The notion of a quasi-free Hilbert module over a function algebra
$\mathcal{A}$ consisting of holomorphic functions on a bounded domain $\Omega$ in complex m
space is introduced. It is shown that quasi-free Hilbert modules correspond to
the completion of the direct sum of a certain number of copies of the algebra
$\mathcal{A}$. A Hilbert module is said to be weakly regular (respectively, regular) if there
exists a module map from a quasi-free module with dense range (respectively,
onto). A Hilbert module $\mathcal{M}$ is said to be compactly supported if there exists a
constant $\beta$ satisfying $\|\varphi f\| \leq \beta \ |\varphi \| \textsl{X} \|f \|$ for some compact subset X of $\Omega$ and
$\varphi$ in $\mathcal{A}$, f in $\mathcal{M}$. It is shown that if a Hilbert module is compactly supported
then it is weakly regular. The paper identifies several other classes of Hilbert
modules which are weakly regular. In addition, this result is extended to yield
topologically exact resolutions of such modules by quasi-free ones. 相似文献
4.
C. J. K. Batty 《Archiv der Mathematik》2003,81(1):72-81
Let $f : \mathbb{R}_{+} \rightarrow \mathbb{C}$ be an exponentially bounded,
measurable function whose Laplace transform has a bounded holomorphic
extension to the open right half-plane. It is known that there is a
constant C such that $\mid \int\limits^t_0 f(s) ds \mid\, \leq C (1 + t)$
for all $t \geq 0$. We show that this estimate is sharp. Furthermore, the
corresponding estimates for orbits of $C_0$-semigroups are also sharp.
Received:17 January 2001; revised manuscropt accepted: 8 February 2001 相似文献
5.
Horst R. Thieme 《Journal of Evolution Equations》2008,8(2):283-305
If T = {T (t); t ≥ 0} is a strongly continuous family of bounded linear operators between two Banach spaces X and Y and f ∈ L
1(0, b, X), the convolution of T with f is defined by . It is shown that T * f is continuously differentiable for all f ∈ C(0, b, X) if and only if T is of bounded semi-variation on [0, b]. Further T * f is continuously differentiable for all f ∈ L
p
(0, b, X) (1 ≤ p < ∞) if and only if T is of bounded semi-p-variation on [0, b] and T(0) = 0. If T is an integrated semigroup with generator A, these respective conditions are necessary and sufficient for the Cauchy problem u′ = Au + f, u(0) = 0, to have integral (or mild) solutions for all f in the respective function vector spaces. A converse is proved to a well-known result by Da Prato and Sinestrari: the generator
A of an integrated semigroup is a Hille-Yosida operator if, for some b > 0, the Cauchy problem has integral solutions for all f ∈ L
1(0, b, X). Integrated semigroups of bounded semi-p-variation are preserved under bounded additive perturbations of their generators and under commutative sums of generators
if one of them generates a C
0-semigroup.
Günter Lumer in memoriam 相似文献
6.
The Sz.-Nagy-FoiaŞ functional model for completely non-unitary contractions is extended to completely non-coisometric sequences
of bounded operatorsT = (T1,...,T
d) (d finite or infinite) on a Hilbert space, with bounded characteristic functions. For this class of sequences, it is shown
that the characteristic function θT is a complete unitary invariant.
We obtain, as the main result, necessary and sufficient conditions for a bounded multi-analytic operator on Fock spaces to
coincide with the characteristic function associated with a completely non-coisometric sequence of bounded operators on a
Hilbert space.
Research supported in part by a COBASE grant from the National Research Council.
The first author was partially supported by a grant from Ministerul Educaţiei Şi Cercetarii.
The second author was partially supported by a National Science Foundation grant. 相似文献
7.
It is shown that for every quasi-normed ideal ${\cal Q}$ of
n-homogeneous continuous polynomials between
Banach spaces there is a quasi-normed ideal ${\cal A}$ of
n-linear continuous mappings ${\cal A}$ such that
$q \in {\cal Q}$ if and only if the associated n-linear
mapping $\check{q}$ of q is in ${\cal A}$.
Received: 12 March 2001 相似文献
8.
T. Hara 《Integral Equations and Operator Theory》1995,23(2):179-204
A bounded linear operatorT is a numerical contraction if and only if there exists a selfadjoint contractionZ such that
. The aim of the present paper is to study the structure of the coreZ(T) of all selfadjoint contractions satisfying the above inequality. Especially we consider several conditions for thatZ(T) is a single-point set. By using this argument we shall characterize extreme points of the set of all numerical contractions. Moreover we shall give effective sufficient conditions for extreme points. 相似文献
9.
We consider the Schrödinger operator H in the space $ L_{2}(\mathbb{R}^{d})$ with
a magnetic potential A(x) decaying as $ \vert x\vert^{-1} $ at infinity and satisfying the
transversal gauge condition <A(x), x > = 0. Our goal is to study properties
of the scattering matrix S() associated to the operator H. In particular,
we find the essential spectrum ess of S() in terms of the behaviour of
A(x) at infinity. It turns out that ess(S()) is normally a rich subset of the
unit circle $\mathbb{T}$ or even coincides with $\mathbb{T}$. We find also the diagonal singularity
of the scattering amplitude (of the kernel of S() regarded as an integral
operator). In general, the singular part S0 of the scattering matrix is a sum
of a multiplication operator and of a singular integral operator. However,
if the magnetic field decreases faster than $ \vert x\vert^{-1} $ for d 3 (and the total
magnetic flux is an integer times 2 for dd = 2), then this singular integral
operator disappears. In this case the scattering amplitude has only a weak
singularity (the diagonal Dirac function is neglected) in the forward direction
and hence scattering is essentially of short-range nature. Moreover, we show
that, under such assumptions, the absolutely continuous parts of the operators
S() and S0 are unitarily equivalent. An important point of our approach is
that we consider S() as a pseudodifferential operator on the unit sphere and
find an explicit expression of its principal symbol in terms of A(x). Another
ingredient is an extensive use (for d 3) of a special gauge adapted to a
magnetic potential A(x). 相似文献
10.
We consider analytic self‐maps φ on $\mathbf {D}$ and prove that the composition operator Cφ acting on $H_{v}^0$ is hypercyclic if φ is an automorphism or a hyperbolic non‐automorphic symbol with no fixed point. We give examples of weights v and parabolic non‐automorphisms φ on $\mathbf {D}$ which yield non‐hypercyclic composition operators Cφ on $H_{v}^0$. 相似文献
11.
Stephan Ramon Garcia 《Integral Equations and Operator Theory》2008,60(3):357-367
If denotes the polar decomposition of a bounded linear operator T, then the Aluthge transform of T is defined to be the operator . In this note we study the relationship between the Aluthge transform and the class of complex symmetric operators (T iscomplex symmetric if there exists a conjugate-linear, isometric involution so that T = CT*C). In this note we prove that: (1) the Aluthge transform of a complex symmetric operator is complex symmetric, (2) if T is complex symmetric, then and are unitarily equivalent, (3) if T is complex symmetric, then if and only if T is normal, (4) if and only if T
2 = 0, and (5) every operator which satisfies T
2 = 0 is necessarily complex symmetric.
This work partially supported by National Science Foundation Grant DMS 0638789. 相似文献
12.
The solution of elementary equations in the Minkowski geometric algebra of complex sets is addressed. For given circular disks
and with radii a and b, a solution of the linear equation
in an unknown set
exists if and only if ab. When it exists, the solution
is generically the region bounded by the inner loop of a Cartesian oval (which may specialize to a limaçon of Pascal, an ellipse, a line segment, or a single point in certain degenerate cases). Furthermore, when a<b<1, the solution of the nonlinear monomial equation
is shown to be the region that is bounded by a single loop of a generalized form of the ovals of Cassini. The latter result is obtained by considering the nth Minkowski root of the region bounded by the inner loop of a Cartesian oval. Preliminary consideration is also given to the problems of solving univariate polynomial equations and multivariate linear equations with complex disk coefficients. 相似文献
13.
14.
Pei Yuan Wu 《Integral Equations and Operator Theory》2006,56(4):559-569
Let A be a bounded linear operator on a complex separable Hilbert space H. We show that A is a C0(N) contraction if and only if
, where U is a singular unitary operator with multiplicity
and x1, . . . , xd are orthonormal vectors satisfying
. For a C0(N) contraction, this gives a complete characterization of its polar decompositions with unitary factors. 相似文献
15.
Mitsuru Uchiyama 《Integral Equations and Operator Theory》2000,37(1):95-105
LetA, B be bounded selfadjoint operators on a Hilbert space. We will give a formula to get the maximum subspace
such that
is invariant forA andB, and
. We will use this to show strong monotonicity or strong convexity of operator functions. We will see that when 0≤A≤B, andB−A is of finite rank,A
t
≤B
t
for somet>1 if and only if the null space ofB−A is invariant forA. 相似文献
16.
In this article, we use a discrete Calderón-type reproducing formula and Plancherel-Pôlya-type inequality associated to a para-accretive function to characterize the Triebel-Lizorkin spaces of para-accretive type $\dot{F}^{\alpha,q}_{b,p}In this article, we use a discrete Calderón-type reproducing formula and Plancherel-P?lya-type inequality associated to a
para-accretive function to characterize the Triebel-Lizorkin spaces of para-accretive type
, which reduces to the classical Triebel-Lizorkin spaces when the para-accretive function is constant. Moreover, we give a
necessary and sufficient condition for the
boundedness of paraproduct operators. From this, we show that a generalized singular integral operator T with M
b
TM
b
∈WBP is bounded from
to
if and only if
and T
*
b=0 for
, where ε is the regularity exponent of the kernel of T.
Chin-Cheng Lin supported by National Science Council, Republic of China under Grant #NSC 97-2115-M-008-021-MY3.
Kunchuan Wang supported by National Science Council, Republic of China under Grant #NSC 97-2115-M-259-009 and NCU Center for
Mathematics and Theoretic Physics. 相似文献
17.
We establish the following Helly-type theorem: Let ${\cal K}$ be a family of
compact sets in $\mathbb{R}^d$. If every d + 1 (not necessarily
distinct) members of ${\cal K}$ intersect in a starshaped set whose kernel
contains a translate of set A, then
$\cap \{ K : K\; \hbox{in}\; {\cal K} \}$ also is a starshaped set whose kernel contains a
translate of A. An analogous result holds
when ${\cal K}$ is a finite family of closed sets in $\mathbb{R}^d$.
Moreover, we have the following planar result: Define function f on
$\{0, 1, 2\}$ by f(0) = f(2) = 3, f(1) = 4. Let ${\cal K}$ be a finite
family of closed sets in the plane. For k = 0, 1, 2, if every f(k)
(not necessarily distinct) members of ${\cal K}$ intersect in a starshaped set
whose kernel has dimension at least k,
then $\cap \{K : K\; \hbox{in}\; {\cal K}\}$ also is a starshaped set whose kernel has
dimension at least k. The number f(k) is best
in each case.Received: 4 June 2002 相似文献
18.
19.
For a Riesz operator T on a reflexive Banach space X with nonzero eigenvalues
denote by E(λi; T) the eigen-projection corresponding to an eigenvalue λi. In this paper we will show that if the operator sequence
is uniformly bounded, then the Riesz operator T can be decomposed into the sum of two operators Tp and Tr: T = Tp + Tr, where Tp is the weak limit of Tn and Tr is quasi-nilpotent. The result is used to obtain an expansion of a Riesz semigroup T(t) for t ≥ τ. As an application, we consider the solution of transport equation on a bounded convex body. 相似文献
20.
Let E and F be idempotent operators on a complex Hilbert space, and let a and b be nonzero scalars with a + b ≠ 0. We prove that aE + bF is Fredholm if and only if E + F is, thus answering affirmatively a question asked by Koliha and Rakočević.
相似文献