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1.
A Toeplitz operator TfT_\phi with symbol f\phi in
L¥(\mathbbD)L^{\infty}({\mathbb{D}}) on the Bergman space
A2(\mathbbD)A^{2}({\mathbb{D}}), where
\mathbbD\mathbb{D} denotes the open unit disc, is radial if f(z) = f(|z|)\phi(z) = \phi(|z|) a.e. on
\mathbbD\mathbb{D}. In this paper, we consider the numerical ranges of such operators. It is shown that all finite line segments, convex hulls
of analytic images of
\mathbbD\mathbb{D} and closed convex polygonal regions in the plane are the numerical ranges of radial Toeplitz operators. On the other hand,
Toeplitz operators TfT_\phi with f\phi harmonic on
\mathbbD\mathbb{D} and continuous on
[`(\mathbbD)]{\overline{\mathbb{D}}} and radial Toeplitz operators are convexoid, but certain compact quasinilpotent Toeplitz operators are not. 相似文献
2.
We study three different problems in the area of Toeplitz operators on the Segal-Bargmann space in Cn. Extending results obtained previously by the first author and Y.L. Lee, and by the second author, we first determine the commutant of a given Toeplitz operator with a radial symbol belonging to the class Sym>0(Cn) of symbols having certain growth at infinity. We then provide explicit examples of zero-products of non-trivial Toeplitz operators. These examples show the essential difference between Toeplitz operators on the Segal-Bargmann space and on the Bergman space over the unit ball. Finally, we discuss the “finite rank problem”. We show that there are no non-trivial rank one Toeplitz operators Tf for f∈Sym>0(Cn). In all these problems, the growth at infinity of the symbols plays a crucial role. 相似文献
3.
An open question, raised independently by several authors, asks if a closed amenable subalgebra of ${\mathfrak{B}(\mathfrak{H})}$ must be similar to an C *-algebra. Recently, Choi, Farah and Ozawa have found a counter-example to this question, but their example is neither separable nor commutative, which leaves the question open for singly-generated algebras. In this paper we continue this line of investigation for special singly-generated algebras. It is shown that if an amenable operator T = N + K, where N is a normal operator, K is a compact operator and σ e (N) has only finite accumulation points, then T is similar to a normal operator; if an amenable operator T = N + K, where N is a normal operator, ${K\in\mathcal{C}_p}$ for some p > 1 and ${\sigma(T)\cup\sigma(N)}$ is included in a smooth Jordan curve, then T is similar to a normal operator; if an amenable operator T = N + Q, where N is a normal operator, Q is a polynomial compact operator and NQ = QN, then T is similar to a normal operator; if there exists p, 1 < p < ∞, such that an amenable operator T satisfies one of the following conditions, then T is similar to a normal operator: (i) ${T-T^*\in\mathcal{C}_p}$ ; (ii) ${I-TT^*\in\mathcal{C}_p}$ ; (iii) ${I-T^*T\in\mathcal{C}_p}$ . 相似文献
4.
Consider j = f +[`(g)]\varphi = f + \overline {g}, where f and g are polynomials, and let TjT_{\varphi} be the Toeplitz operators with the symbol j\varphi. It is known that if TjT_{\varphi} is hyponormal then |f¢(z)|2 3 |g¢(z)|2|f'(z)|^{2} \geq |g'(z)|^{2} on the unit circle in the complex plane. In this paper, we show that it is also a necessary and sufficient condition under
certain assumptions. Furthermore, we present some necessary conditions for the hyponormality of TjT_{\varphi} on the weighted Bergman space, which generalize the results of I. S. Hwang and J. Lee. 相似文献
5.
Jong-Do Park 《Integral Equations and Operator Theory》2006,54(4):571-584
We investigate necessary and sufficient conditions for boundedness of the operator
on the Bergman space of the unit ball
for n ≥ 1, where Tf is the Toeplitz operator. Those conditions are related to boundedness of the Berezin transform of symbols f and g. We construct the inner product formula which plays a crucial role in proving the sufficiency of the conditions. 相似文献
6.
7.
We show that T is a surjective multiplicative (but not necessarily linear) isometry from the Smirnov class on the open unit disk, the ball, or the polydisk onto itself,
if and only if there exists a holomorphic automorphism Φ such that T(f)=f ○ Φ for every class element f or T(f) = [`(f° [`(j)] )]\overline {f^\circ \bar \varphi } for every class element f, where the automorphism Φ is a unitary transformation in the case of the ball and Φ(z
1, ..., z
n
) = (l1 zi1 ,...,ln zin )(\lambda _1 z_{i_1 } ,...,\lambda _n z_{i_n } ) for |λ
j
| = 1, 1 ≤ j ≤ n, and (i
1; ..., i
n
)is some permutation of the integers from 1through n in the case of the n-dimensional polydisk. 相似文献
8.
Caixing Gu 《Journal of Functional Analysis》2004,215(1):178-205
Let X be a bounded linear operator on the Hardy space H2 of the unit disk. We show that if is of finite rank for every inner function θ, then X=T?+F for some Toeplitz operator T? and some finite rank operator F on H2. This solves a variant of an open question where the compactness replaces the finite rank conditions. 相似文献
9.
Trieu Le 《Integral Equations and Operator Theory》2010,67(3):425-438
Let ${\vartheta}$ be a measure on the polydisc ${\mathbb{D}^n}$ which is the product of n regular Borel probability measures so that ${\vartheta([r,1)^n\times\mathbb{T}^n) >0 }$ for all 0 < r < 1. The Bergman space ${A^2_{\vartheta}}$ consists of all holomorphic functions that are square integrable with respect to ${\vartheta}$ . In one dimension, it is well known that if f is continuous on the closed disc ${\overline{\mathbb{D}}}$ , then the Hankel operator H f is compact on ${A^2_\vartheta}$ . In this paper we show that for n ≥ 2 and f a continuous function on ${{\overline{\mathbb{D}}}^n}$ , H f is compact on ${A^2_\vartheta}$ if and only if there is a decomposition f = h + g, where h belongs to ${A^2_\vartheta}$ and ${\lim_{z\to\partial\mathbb{D}^n}g(z)=0}$ . 相似文献
10.
In this paper we consider APN functions ${f:\mathcal{F}_{2^m}\to \mathcal{F}_{2^m}}$ of the form f(x) = x ?1 + g(x) where g is any non ${\mathcal{F}_{2}}$ -affine polynomial. We prove a lower bound on the degree of the polynomial g. This bound in particular implies that such a function f is APN on at most a finite number of fields ${\mathcal{F}_{2^m}}$ . Furthermore we prove that when the degree of g is less than 7 such functions are APN only if m ?? 3 where these functions are equivalent to x 3. 相似文献
11.
Ekaterina Shulman 《Aequationes Mathematicae》2010,79(1-2):13-21
Let a representation T of a unital topological semigroup G on a topological linear space X be given. We call ${x \in X}$ a finite vector if its orbit T(G)x is contained in a finite dimensional subspace. In this paper some statements on finite vectors will be proved and applied to the functional equation $$ f(g_1g_2\cdots g_n) = \sum_{E}\sum_{j=1}^{N_E}u^E_jv^E_j $$ where E runs through all proper non-empty subsets of ${\{1,2,\ldots,n\}, N_E \in \mathbb{N}}$ , and for each E, the functions ${u^E_j}$ only depend on variables g i with ${i\in E}$ , while the ${v^E_j}$ only depend on g i with ${i\notin E}$ . 相似文献
12.
Let and suppose that f : K
n
→K
n
is nonexpansive with respect to the l
1-norm, , and satisfies f (0) = 0. Let P
3(n) denote the (finite) set of positive integers p such that there exists f as above and a periodic point of f of minimal period p. For each n≥ 1 we use the concept of 'admissible arrays on n symbols' to define a set of positive integers Q(n) which is determined solely by number theoretical and combinatorial constraints and whose computation reduces to a finite
problem. In a separate paper the sets Q(n) have been explicitly determined for 1 ≤n≤ 50, and we provide this information in an appendix. In our main theorem (Theorem 3.1) we prove that P
3(n) = Q(n) for all n≥ 1. We also prove that the set Q(n) and the concept of admissible arrays are intimately connected to the set of periodic points of other classes of nonlinear
maps, in particular to periodic points of maps g : D
g→D
g, where is a lattice (or lower semilattice) and g is a lattice (or lower semilattice) homomorphism. 相似文献
13.
Nadia J. Gal James E. Jamison Aristomenis G. Siskakis 《Complex Analysis and Operator Theory》2010,4(2):245-255
We are interested in the isometric equivalence problem for the Cesàro operator
C(f) (z) = \frac1z ò0zf(x) \frac11-xd x{C(f) (z) =\frac{1}{z} \int_{0}^{z}f(\xi) \frac{1}{1-\xi}d \xi} and an operator
Tg(f)(z)=\frac1zò0zf(x) g¢(x) d x{T_{g}(f)(z)=\frac{1}{z}\int_{0}^{z}f(\xi) g^{\prime}(\xi) d \xi}, where g is an analytic function on the disc, on the Hardy and Bergman spaces. Then we generalize this to the isometric equivalence
problem of two operators Tg1{T_{g_{1}}} and Tg2{T_{g_{2}}} on the Hardy space and Bergman space. We show that the operators Tg1{T_{g_{1}}} and Tg2{T_{g_{2}}} satisfy Tg1U1=U2Tg2{T_{g_{1}}U_{1}=U_{2}T_{g_{2}}} on H
p
, 1 ≤ p < ∞, p ≠ 2 if and only if g2(z) = lg1(eiqz){g_{2}(z) =\lambda g_{1}(e^{i\theta}z) }, where λ is a modulus one constant and U
i
, i = 1, 2 are surjective isometries of the Hardy Space. This is analogous to the Campbell-Wright result on isometrically equivalence
of composition operators on the Hardy space. 相似文献
14.
Suppose that f : [0, 1] ?? [0, 2] is a continuous strictly increasing piecewise differentiable function, and define T f x :=?f(x) (mod 1). Let ${\beta \geq \sqrt[3]{2}}$ . It is proved that T f is topologically transitive if inf f???????? and ${f(0)\geq\frac{1}{\beta+1}}$ . Counterexamples are provided if the assumptions are not satisfied. For ${\sqrt[3]{2}\leq\beta < \sqrt{2}}$ and 0????????? 2 ? ?? it is shown that ??x?+??? (mod 1) is topologically transitive if and only if ${\alpha < \frac{1}{\beta^2+\beta}}$ or ${\alpha >2 -\beta-\frac{1}{\beta^2+\beta}}$ . 相似文献
15.
Egor A. Alekhno 《Integral Equations and Operator Theory》2010,67(3):301-326
For the lower Weyl spectrum $$\sigma_{\rm w}^-(T) = \bigcap_{0 \le K \in \mathcal{K}(E) \le T} \sigma(T - K),$$ where T is a positive operator on a Banach lattice E, the conditions for which the equality ${\sigma_{\rm w}^-(T) = \sigma_{\rm w}^-(T^*)}$ holds, are established. In particular, it is true if E has order continuous norm. An example of a weakly compact positive operator T on ? ∞ such that the spectral radius ${r(T) \in \sigma_{\rm w}^-(T) {\setminus} (\sigma_{\rm f}(T) \cup \sigma_{\rm w}^-(T^*))}$ , where σ f(T) is the Fredholm spectrum, is given. The conditions which guarantee the order continuity of the residue T ?1 of the resolvent R(., T) of an order continuous operator T ≥ 0 at ${r(T) \notin \sigma_{\rm f}(T)}$ , are discussed. For example, it is true if T is o-weakly compact. It follows from the proven results that a Banach lattice E admitting an order continuous operator T ≥ 0, ${r(T) \notin \sigma_{\rm f}(T)}$ , can not have the trivial band ${E_n^\sim}$ of order continuous functionals in general. It is obtained that a non-zero order continuous operator T : E → F can not be approximated in the r-norm by the operators from ${E_\sigma^\sim \otimes F}$ , where F is a Banach lattice, ${E_\sigma^\sim}$ is a disjoint complement of the band ${E_n^\sim}$ of E*. 相似文献
16.
In 1999 Nina Zorboska and in 2003 P. S.Bourdon, D. Levi, S.K.Narayan and J.H. Shapiro investigated the essentially normal composition operator ${C_\varphi }$ , when φ is a linear-fractional self-map of D. In this paper first, we investigate the essential normality problem for the operator T w ${C_\varphi }$ on the Hardy space H 2, where w is a bounded measurable function on ?D which is continuous at each point of F(φ), φ ∈ S(2), and T w is the Toeplitz operator with symbol w. Then we use these results and characterize the essentially normal finite linear combinations of certain linear-fractional composition operators on H 2. 相似文献
17.
The aim of this work is to show that in any complete Riemannian
manifold M, without boundary, the curvature operator is nonnegative
if and only if the Dirac Laplacian D2 generates a C*-Markovian
semigroup (i.e. a strongly continuous, completely positive, contraction
semigroup) on the Cliord C*-algebra of Mor, equivalently, if
and only if the quadratic form $\mathcal{E}$D of D2
is a C*-Dirichlet form. 相似文献
18.
Nik Stopar 《Semigroup Forum》2012,85(2):322-336
In this paper we investigate the ascending chain conditions on principal left and right ideals for semidirect products of semigroups and show how this is connected to the corresponding problem for rings of skew generalized power series. Let S be a left cancellative semigroup with a unique idempotent e, T a right cancellative semigroup with an idempotent f and $\omega: T \to \operatorname {End}(S)$ a semigroup homomorphism such that ??(f)=id S . We show that in this case the semidirect product S? ?? T satisfies the ascending chain condition for principal left ideals (resp. right ideals) if and only if S and T satisfy the ascending chain condition for principal left ideals (resp. right ideals and $\operatorname {Im}\omega(t)$ is closed for complete inverses for all t??T). We also give several examples to show that for more general semigroups these implications may not hold. 相似文献
19.
Tim Netzer 《Journal of Geometric Analysis》2010,20(3):751-770
We consider a closed set S?? n and a linear operator that preserves nonnegative polynomials, in the following sense: if f≥0 on S, then Φ(f)≥0 on S as well. We show that each such operator is given by integration with respect to a measure taking nonnegative functions as its values. This can be seen as a generalization of Haviland’s Theorem, which concerns linear functionals on ?[X 1,…,X n ]. For compact sets S we use the result to show that any nonnegativity preserving operator is a pointwise limit of very simple nonnegativity preservers with finite dimensional range.
相似文献
$\Phi \colon \mathbb{R}[X_1,\ldots,X_n]\rightarrow \mathbb{R}[X_1,\ldots,X_n]$
20.
Square matrices of the form ${X_n = T_n + f_n(T_n^{-1})^*}$ , where T n is a ${n \times n}$ invertible banded Toeplitz matrix and f n some positive sequence are considered. Convergence via an order estimate is proven for the difference of ${\|X_n^{-1}\|}$ and a function depending only on f n . Fredholmness of the infinite counterpart of T n is shown to greatly affect this result. A correction of a proof in the paper on which the current research is based, is appended as well. 相似文献