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.

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

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.

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.

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.

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.

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.

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.

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.

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.