Spelling suggestions: "subject:"toeplitz operator"" "subject:"teplitz operator""
11 |
Invertibility of a Class of Toeplitz Operators over the Half PlaneVasilyev, Vladimir 28 September 2006 (has links)
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.
|
12 |
On infinite matrices whose entries satisfying certain dyadic recurrent formulaHsu, Chia-ming 25 July 2007 (has links)
Let (b$_{i,j}$) be a bounded matrix on extit{ l}$^{2}$, $Bbb
T={zinBbb C:|z|=1}$, and A be a bounded matrix on L$^{
2}(mathbb{T)}$ satisfying the conditions
1.$langle Az^{2j},z^{2i}
angle =sigma ^{-1}b_{ij}+|alpha
|^{2}sigma ^{-1}langle Az^{j},z^{i}
angle $;
2.$langle Az^{2j},z^{2i-1}
angle =-alpha sigma
^{-1}b_{ij}+alpha sigma ^{-1}langle Az^{j},z^{i}
angle $;
3.$langle Az^{2j-1},z^{2i}
angle =-overline{alpha }sigma
^{-1}b_{ij}+overline{alpha }sigma ^{-1}langle
Az^{j},z^{i}
angle$;
4.$langle Az^{2j-1},z^{2i-1}
angle =|alpha |^{2}sigma
^{-1}b_{ij}+sigma ^{-1}langle Az^{j},z^{i}
angle $
hspace{-0.76cm} for all $i,jin mathbb{Z}$, where $sigma
=1+|alpha |^{2},,alpha in mathbb{C},alpha
eq0$.
The above conditions evidently suggests that there is a "dyadic"
relation in the entries of $A$. Here in the following picture
illustrates how each $ij-$th entry of $A$ generates the 2 by 2 block
in $A$ with entries ${a_{2i 2j}, a_{2i-1 2j}, a_{2i 2j-1},
a_{2i-1 2j-1}}.$ vspace{-0.3cm}
egin{figure}[hp]
egin{center}
includegraphics[scale=0.42]{cubic.pdf}
end{center}
vspace{-0.8cm}caption{The dyadic recurrent form} end{figure}
It has been shown [2] that $displaystyle A=sum_{n=0}^{infty
}S^{n}BS^{ast n}$, where $Sz^i=sigma ^{-1/2}(overline{alpha
}z^{2i}+z^{2i-1})$ and $$B=sumlimits_{i=-infty}^infty
sumlimits_{j=-infty}^infty b_{ij}(u_{i}otimes u_{j}),
u_{i}(z)=sigma ^{-1/2}z^{2i-1}(alpha -z).$$
In this paper, we shall use the above relations to compute $langle
a_{i,j}
angle $ explicitly.
ewline
Key words: shift operator, bounded matrix, dyadic recurrent formula,
slant Toeplitz operator, separable Hilbert space
2.$langle Az^{2j},z^{2i-1}
angle =-alpha sigma
^{-1}b_{ij}+alpha sigma ^{-1}langle Az^{j},z^{i}
angle $
3.$langle Az^{2j-1},z^{2i}
angle =-overline{alpha }sigma
^{-1}b_{ij}+overline{alpha }sigma ^{-1}langle
Az^{j},z^{i}
angle $
4.$langle Az^{2j-1},z^{2i-1}
angle =|alpha |^{2}sigma
^{-1}b_{ij}+sigma ^{-1}langle Az^{j},z^{i}
angle $
for all $i,jin mathbb{Z}$, where $sigma =1+|alpha
|^{2},,alpha in mathbb{C},alpha
eq0$
egin{figure}[hp]
egin{center}
includegraphics[scale=0.42]{cubic.pdf}
end{center}
caption{The dyadic recurrent form} end{figure}
Since it has been
shown [2] that $displaystyle A=sum_{n=0}^{infty }S^{n}BS^{ast
n}$, where
$ Sz^i=sigma ^{-1/2}(overline{alpha }z^{2i}+z^{2i-1})$
$ B=sum sum b_{ij}(u_{i}otimes u_{j})$ ;;; which
$u_{i}(z)=sigma ^{-1/2}z^{2i-1}(alpha -z)$
Then we can use it to compute $langle Az^{j},z^{i}
angle $
explicity if A satisfies the previous condition.
ewline
Key words: shift operator, bounded matrix, dyadic recurrent formula,
slant Toeplitz operator, separable Hilbert space
|
13 |
Estimates for the condition numbers of large semi-definite Toeplitz matricesBöttcher, A., Grudsky, S. M. 30 October 1998 (has links) (PDF)
This paper is devoted to asymptotic estimates for the condition numbers
$\kappa(T_n(a))=||T_n(a)|| ||T_n^(-1)(a)||$
of large $n\cross n$ Toeplitz matrices $T_N(a)$ in the case where
$\alpha \element L^\infinity$ and $Re \alpha \ge 0$ . We describe several classes
of symbols $\alpha$ for which $\kappa(T_n(a))$ increases like $(log n)^\alpha, n^\alpha$ ,
or even $e^(\alpha n)$ . The consequences of the results for singular values, eigenvalues,
and the finite section method are discussed. We also consider Wiener-Hopf integral
operators and multidimensional Toeplitz operators.
|
14 |
Factorization theory for Toeplitz plus Hankel operators and singular integral operators with flipEhrhardt, Torsten 02 September 2004 (has links) (PDF)
In this habilitation thesis a factorization theory for Toeplitz plus Hankel operators and singular integral operators with flip is established. These operators are considered with matrix-valued symbols and are thought of acting on the vector-valued analogues of the Hardy and Lebesgue spaces.
A factorization theory for pure Toeplitz operators and singular integral operators without flip is known since decades and provides necessary and sufficient conditions for Fredholmness and formulas for the defect numbers. In particular, the invertibility of such operators is equivalent to the existence of a certain type of Wiener-Hopf factorization.
In this thesis an analogous theory for the afore-mentioned more general classes of operators is developed. It turns out that a completely different kind of factorization is needed. This kind of factorization is studied extensively, and a corresponding Fredholm theory is established. A connection with the Hunt-Muckenhoupt-Wheeden condition is made, and several examples and applications are given as well. / In dieser Habilitationsschrift wird eine Faktorisierungstheorie für Toeplitz plus Hankel-Operatoren und singuläre Integraloperatoren mit Flip aufgestellt. Diese Operatoren werden mit matrixwertigem Symbol betrachtet und sind auf den vektorwertigen Analoga der Hardy- und Lebesgue-Räumen definiert.
Eine Faktorisierungstheorie für reine Toeplitz bzw. singuläre Integraloperatoren ohne Flip ist seit Jahrzehnten bekannt. Sie liefert notwendige und hinreichende Bedingungen für die Fredholmeigenschaft und Formeln für die Defektzahlen. Insbesondere ist die Invertierbarkeit derartiger Operatoren äquivalent zur Existenz einer bestimmten Art der Wiener-Hopf-Faktorisierung.
In dieser Habilitationsschrift wird eine entsprechende Theorie für die erwähnten, allgemeineren Klassen von Operatoren aufgestellt. Es stellt sich heraus, dass eine völlig andere Art der Faktorisierung benötigt wird. Diese Art der Faktorisierung wird eingehend studiert und eine entsprechende Fredholmtheorie wird entwickelt. Ein Zusammenhang mit der Hunt-Muckenhoupt-Wheeden Bedingung wird hergestellt. Mehrere Beispiele und Anwendungen werden ebenfalls angegeben.
|
15 |
Generalized convolution operators and asymptotic spectral theoryZabroda, Olga Nikolaievna 14 December 2006 (has links) (PDF)
The dissertation contributes to the further advancement of the theory of various classes of discrete and continuous (integral) convolution operators. The thesis is devoted to the study of sequences of matrices or operators which are built up in special ways from generalized discrete or continuous convolution operators. The generating functions depend on three variables and this leads to considerably more complicated approximation sequences. The aim was to obtain for each case a result analogous to the first Szegö limit theorem providing the first order asymptotic formula for the spectra of regular convolutions.
|
16 |
Über die Splitting-Eigenschaft der Approximationszahlen von Matrix-Folgen: l1-TheorieSeidel, Markus 02 February 2007 (has links) (PDF)
In dieser Arbeit wird das asymptotische Verhalten der Approximationszahlen für Operatorfolgen aus einer speziellen Klasse von Banachalgebren untersucht. Es werden bemerkenswerte Eigenschaften der Folgen und der Approximationszahlen ihrer Operatoren gezeigt, darunter die so genannte splitting-Eigenschaft.
Ein typisches Beispiel solcher Operatorfolgen stellen die Finite Sections von Toeplitzoperatoren dar, die exemplarisch behandelt werden. Dabei werden hier auch die Folgenräume l1 und l-unendlich betrachtet.
|
17 |
Generalized convolution operators and asymptotic spectral theoryZabroda, Olga Nikolaievna 11 December 2006 (has links)
The dissertation contributes to the further advancement of the theory of various classes of discrete and continuous (integral) convolution operators. The thesis is devoted to the study of sequences of matrices or operators which are built up in special ways from generalized discrete or continuous convolution operators. The generating functions depend on three variables and this leads to considerably more complicated approximation sequences. The aim was to obtain for each case a result analogous to the first Szegö limit theorem providing the first order asymptotic formula for the spectra of regular convolutions.
|
18 |
Über die Splitting-Eigenschaft der Approximationszahlen von Matrix-Folgen: l1-TheorieSeidel, Markus 16 January 2006 (has links)
In dieser Arbeit wird das asymptotische Verhalten der Approximationszahlen für Operatorfolgen aus einer speziellen Klasse von Banachalgebren untersucht. Es werden bemerkenswerte Eigenschaften der Folgen und der Approximationszahlen ihrer Operatoren gezeigt, darunter die so genannte splitting-Eigenschaft.
Ein typisches Beispiel solcher Operatorfolgen stellen die Finite Sections von Toeplitzoperatoren dar, die exemplarisch behandelt werden. Dabei werden hier auch die Folgenräume l1 und l-unendlich betrachtet.
|
19 |
Second-Order Trace Formulas in Szegö-type TheoremsVasilyev, Vladimir 15 February 2007 (has links) (PDF)
A new way of proof of Szegö-type theorems is
presented. The idea of the proof is based on the
construction of "almost" inverse operator to
the finite section T_n(a) of a Toeplitz operator T(a),
which is close to the inverse operator in the trace
norm (these "almost" inverses are well-known).
This way of proof gives the possibility to write
another representation for the second constant
E_f(a), and in the scalar case to receive a
shorter representation. Another observation is
that the convergence in these theorems is
strongly dependent on the smoothness of the
generating function a.
|
20 |
Estimates for the condition numbers of large semi-definite Toeplitz matricesBöttcher, A., Grudsky, S. M. 30 October 1998 (has links)
This paper is devoted to asymptotic estimates for the condition numbers
$\kappa(T_n(a))=||T_n(a)|| ||T_n^(-1)(a)||$
of large $n\cross n$ Toeplitz matrices $T_N(a)$ in the case where
$\alpha \element L^\infinity$ and $Re \alpha \ge 0$ . We describe several classes
of symbols $\alpha$ for which $\kappa(T_n(a))$ increases like $(log n)^\alpha, n^\alpha$ ,
or even $e^(\alpha n)$ . The consequences of the results for singular values, eigenvalues,
and the finite section method are discussed. We also consider Wiener-Hopf integral
operators and multidimensional Toeplitz operators.
|
Page generated in 0.0525 seconds