Spelling suggestions: "subject:"toeplitz operator"" "subject:"teplitz operator""
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A simple comparison between the Toeplitz and the £f -Toeplitz operatorsLi, Chieh-cheng 09 July 2009 (has links)
Let £f be a complex number in the closed unit disc, and H be a separable Hilbert space with the orthonormal basis, say, £`={e_n:n=0,1,2,¡K}. A bounded operator T on H is called a £f-Toeplitz operator if < Te_{m+1},Te_{n+1} >=£f< Te_m,Te_n > (where <¡E,¡E> is the inner product on H).
The subject arises just recently from a special case of the operator equation S*AS = £fA + B, where S is a shift on H, which plays an essential role in finding bounded matrix (a_{ij}) on l^2(Z) that solves the system of equations
a_{2i,2j} = p_{ij} + aa_{ij}
a_{2i,2j−1} = q_{ij} + ba_{ij}
a_{2i−1,2j} = v_{ij} + ca_{ij}
a_{2i−1,2j−1} = w_{ij} + da_{ij}
for all i, j ∈ Z, where (p_{ij}), (q_{ij}), (v_{ij}), (w_{ij}) are bounded matrices on l^2(Z) and a, b, c, d ∈C.
It is also clear that the well-known Toeplitz operators are precisely the solutions of S*AS = A, when S is the unilateral shift. The purpose of this paper is to discuss some basic topics, such as boundedness and compactness, of the £f-Toeplitz operators, and study the similarities and the differences with the corresponding results for the Toeplitz operators.
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Fredholm spectra of £f-Toeplitz operatorsChen, Chih-Hao 25 July 2011 (has links)
Abstract
Let £f be a complex number in the closed unit disc , and H be a separable Hilbert space with the orthonormal basis, say,£`= {en : n =0 , 1 , 2¡K}. A bounded operator T on H is called a £f-Toeplitz operator if <Tem+1 , en+1> =£f <Tem , en> (where <¡E,¡E> is the inner product on H).If the function £p can be represented as a linear combination of the above orthonormal basis with the coefficients an=<Te0 ,en >, n≥ 0,and an=<Telnl ,e0 >, n<0, then we call this the symbol of T . The subject arises naturally from a special case of the operator equation
S*AS =£fA + B; where S is a shift on H ,
and in this operator equation the matrix A can solve a special set of simultaneous equations.
It is also clear that the well-known Toeplitz operators are precisely the solutions of S*AS = A, when S is the unilateral shift.In this paper,we will review the similarities and differences between £f-Toeplitz operators and Toeplitz operators. The main purpose is to generalize the well-known Coburn's characterization for the essential spectrum(or,the same in this case,spectrum)for Toeplitz operators to £f-Toeplitz operators.
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Über die Splitting-Eigenschaft der Approximationszahlen von Matrix-Folgen : l1-Theorie$nElektronische Ressource /Seidel, Markus, Silbermann, Bernd. January 2006 (has links)
Chemnitz, Techn. Univ., Diplomarb., 2006.
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Operators which are constant with respect to slant Toeplitz operatorsChen, Chien-chou 04 July 2006 (has links)
Let H be a separable Hilbert space and {e_n : n belong to Z} be an orthonormal basis in H. A bounded operator T is called the slant Toeplitz operator if <T ej , ei> =c2i−j , where c_n is the n-th Fourier series of a bounded Lebesgue measurable function on the unit circle T = {z belong to C : |z| = 1}. It has been shown [7] that T* is an isometry if and only if |fi(z)|^2 +|fi(−z)|^2 = 2 a.e. on T and if this is the case and fi belong to C(T), then either T is unitarily equivalent to a shift or to the direct sum of a shift and a rank one unitary, with infinite multiplicity
(for the shift part, that is). Moreover, with some additional assumption on the smoothness and the zeros of fi, T* is similar to either the constant multiple of a shift or to the constant multiple of the direct sum of a shift and a rank one unitary, with infinite multiplicity. On the other hand, according to the terminologies in [10], an operator A that is constant with respect to a shift S if AS = SA and A S = SA . Therefore, in this article, we will study the operators that is constant with respect to T , i.e., bounded operator A satisfying AT = T A and A T = T A .
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Invertibility of a class of Toeplitz operators over the half planeVasil'ev, Vladimir A., January 2007 (has links)
Chemnitz, Techn. Univ., Diss., 2007.
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Structure of Toeplitz-composition operatorsSyu, Meng-Syun 14 February 2011 (has links)
Let $vp$ be a $L^infty$ function on the unit circle $Bbb T$ and
$ au$ be an elliptic automorphism on the unit disc $Bbb D$. In this paper, we will show that $T_vp C_ au$, i.e., the product of the Toeplitz operator $T_vp$ and the composition operator $C_ au$ on $H^2$, is similar to a block Toeplitz matrix if $ au$ has finite
order.
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Eigenvectors for Certain Action on B(H) Induced by ShiftCheng, Rong-Hang 05 September 2011 (has links)
Let $l^2(Bbb Z)$ be the Hilbert space of square summable double sequences of complex numbers with standard basis ${e_n:ninBbb Z}$, and let us consider a bounded matrix $A$ on $l^2(Bbb Z)$
satisfying the following system of equations
egin{itemize}
item[1.] $lan
Ae_{2j},e_{2i}
an=p_{ij}+alan Ae_{j},e_i
an$;
item[2.] $lan
Ae_{2j},e_{2i-1}
an=q_{ij}+blan Ae_{j},e_{i}
an$;
item[3.] $lan
Ae_{2j-1},e_{2i}
an=v_{ij}+clan Ae_{j},e_{i}
an$;
item[4.] $lan
Ae_{2j-1},e_{2i-1}
an=w_{ij}+dlan Ae_{j},e_{i}
an$
end{itemize}
for all $i,j$, where $P=(p_{ij})$, $Q=(q_{ij})$, $V=(v_{ij})$, $W=(w_{ij})$ are bounded matrices on $l^2(Bbb Z)$ and $a,b,c,dinBbb C$. This type dyadic recurrent system arises in the study of bounded operators commuting with the slant Toeplitz operators, i.e., the class of operators ${{cal T}_vp:vpin L^infty(Bbb T)}$ satisfying $lan {cal T}_vp e_j,e_i
an=c_{2i-j}$, where $c_n$ is the $n$-th Fourier coefficient of $vp$.
It is shown in [10] that the solutions of the above system are closely related to the bounded solution $A$ for the operator equation
[
phi(A)=S^*AS=lambda A+B,
]
where $B$ is fixed, $lambdainBbb C$ and $S$ the shift given by ${cal T}_{arzeta+arxi z}^*$ (with $zetaxi
ot=0$ and $|zeta|^2+|xi|^2=1$). In this paper, we shall characterize the ``eigenvectors" for $phi$ for the eigenvalue $lambda$ with
$|lambda|leq1$, in terms of dyadic recurrent systems similar to the one above.
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Spaces of Analytic Functions and Their ApplicationsMitkovski, Mishko 2010 August 1900 (has links)
In this dissertation we consider several problems in classical complex analysis and operator theory. In the first part we study basis properties of a system of complex exponentials with a given frequency sequence. We show that most of these basis properties can be characterized in terms of the invertibility properties of certain Toeplitz operators. We use this reformulation to give a metric description of the radius of l2-dependence. Using similar methods we solve the classical Beurling gap problem in the case of separated real sequences. In the second part we consider the classical Polýa-Levinson problem asking for a description of all real sequences with the property that every zero type entire function which is bounded on such a sequence must be a constant function. We first give a description in terms of injectivity of certain Toeplitz operators and then use this to give a metric description of all such sequences. In the last part we study the spectral changes of a partial isometry under unitary perturbations. We show that all the spectra can be described in terms of the characteristic function of the partial isometry that is being perturbed. Our main tool in the proofs is a Herglotz-type representation for generalized spectral measures. We furthermore use this representation to give a new proof of the classical Naimark's dilation theorem and to generalize Aleksandrov's disintegration theorem.
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Second-order trace formulas in Szegö-type theoremsVasil'ev, Vladimir A., Silbermann, Bernd. January 2007 (has links)
Chemnitz, Techn. Univ., Masterarb., 2002.
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Invertibility of a Class of Toeplitz Operators over the Half PlaneVasilyev, Vladimir 07 February 2007 (has links) (PDF)
This dissertation is concerned with invertibility and
one-sided invertibility of Toeplitz operators
over the half plane whose generating functions
admit homogenous discontinuities, and with
stability of their pseudo finite sections.
The invertibility criterium is given in terms
of invertibility of a family of one
dimensional Toeplitz operators with piecewise
continuous generating functions. The one-sided
invertibility criterium is given it terms of
constraints on the partial indices of certain
Toeplitz operator valued function.
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