A maximum problem for matrices

The following maximum problem is considered: To find among all contractions T on an n-dimensional Hilbert space whose spectral radius does not exceed a given number p< 1, the operator T for which |Tⁿ| is maximum. A matrix T of Toeplitz type is constructed for which this maximum is attained.     

A direct method to solve block banded block Toeplitz systems with non-banded Toeplitz blocks

A fast solution algorithm is proposed for solving block banded block Toeplitz systems with non-banded Toeplitz blocks. The algorithm constructs the circulant transformation of a given Toeplitz system and then by means of the Sherman-Morrison-Woodbury formula transforms its inverse to an inverse of the original matrix. The block circulant matrix with Toeplitz blocks is converted to a block diagonal matrix with Toeplitz blocks, and the resulting Toeplitz systems are solved by means of a fast Toeplitz solver.The computational complexity in the case one uses fast Toeplitz solvers is equal to ξ(m,n,k)=O(mn³)+O(k³n³) flops, there are m block rows and m block columns in the matrix, n is the order of blocks, 2k+1 is the bandwidth. The validity of the approach is illustrated by numerical experiments.     

A continuation method for solving symmetric Toeplitz systems

A fast algorithm is proposed for solving symmetric Toeplitz systems. This algorithm continuously transforms the identity matrix into the inverse of a given Toeplitz matrix T. The memory requirements for the algorithm are O(n), and its complexity is O(log κ(T)nlogn), where (T) is the condition number of T. Numerical results are presented that confirm the efficiency of the proposed algorithm.     

On the frequency of titchmarsh’s phenomenon for ζ(s)—III

We obtain a lower bound for max |ζ(1/2+it)| ast varies overT?t?T+Y, where (logT)^1/100?Y?T, as a function ofY(1/100 is unimportant). Our lower bound is exp {D(logY)^1/2 (log logY)^?1/2} whereD is a positive constant. (After submitting this paper for publication we came to know through a preprint of H L Montgomery that he had proved our result in the caseY=T. In his proof an essential assumption is Riemann hypothesis and our result is independent of any such unproved hypothesis. However he has other new results which are free from any hypothesis).     

Solution of systems with Toeplitz matrices generated by rational functions

We consider Toeplitz matrices T_n = (t_i−j)ⁿ_i,j=0, where Σ^∞_−∞t_jz^j is a formal Laurent series of a rational function R(z). A criterion is given for T_n to be invertible, in terms of the nonvanishing of a determinant D_n involving the zeros of R(z), and of order and form independent of n; i.e., n enters into D_n as a parameter, and not so as to complicate D_n as n increases. Explicit formulas involving similar determinants are given for the solution of the system T_nX = Y in the case where T_n is invertible. Formulas are also given for T⁻¹_n in the case where T_n−1 and T_n are both convertible Suggestions concerning possible computational procedures based on the results are included.     

Toeplitz operators on Bergman spaces

Let ? be an element in \(H^\infty (D) + C(\overline D )\) such that ?^* is locally sectorial. In this paper it is shown that the Toeplitz operator defined on the Bergman spaceA ² (D) is Fredholm. Also, it is proved that ifS is an operator onA ²(D) which commutes with the Toeplitz operatorT _? whose symbol ? is a finite Blaschke product, thenS H ^∞ (D) is contained inH ^∞ (D). Moreover, some spectral properties of Toeplitz operators are discussed, and it is shown that the spectrum of a class of Toeplitz operators defined on the Bergman spaceA ² (D), is not connected.     

Asymptotic behaviours and generalized Toeplitz operators

We study some generalized Toeplitz operators associated to operators T on a Hilbert space H, for which there exists the limit of {‖Tnh‖} for every h∈H. We refer to the asymptotic limit ST of such a T, in the sense of [L. Kerchy, Operators with regular norm-sequences, Acta Sci. Math. (Szeged) 63 (1997) 571-605; L. Kerchy, Generalized Toeplitz operators, Acta Sci. Math. (Szeged) 68 (2002) 373-400; G. Cassier, Generalized Toeplitz operators, restrictions to invariant subspaces and similarity problems, J. Operator Theory 53 (1) (2005) 101-140; C.S. Kubrusly, An Introduction to Models and Decompositions in Operator Theory, Birkhäuser, Boston, 1997], and we give some conditions of ergodicity for T. Also, certain results of Douglas [R.G. Douglas, On the operator equation S^∗XT=X and related topics, Acta Sci. Math. (Szeged) 30 (1969) 19-32] involving generalized Toeplitz operators are extended in our more general setting, and we apply these results to ρ-contractions.     

Inversion of Toeplitz operators,Levinson equations,and Gohberg-Krein factorization—A simple and unified approach for the rational case

The problem of the inversion of the Toeplitz operator T_Φ, associated with the operator-valued function Φ defined on the unit circle, is known to involve the associated Levinson system of equations and the Gohberg-Krein factorization of Φ. A simplified and self-contained approach, making clear the connections between these three problems, is presented in the case where Φ is matrix-valued and rational. The key idea consists in looking at the Levinson system of equations associated with Φ^?1(z^?1), rather than that associated with Φ(z). As a consequence, a new invertibility criterion for Toeplitz operators with rational matrix-valued symbols is derived.     

Fast solution of toeplitz systems of equations and computation of Padé approximants

We present two new algorithms, ADT and MDT, for solving order-n Toeplitz systems of linear equations Tz = b in time O(n log²n) and space O(n). The fastest algorithms previously known, such as Trench's algorithm, require time $\text{Ω(n}{}^2\text{)}$ and require that all principal submatrices of T be nonsingular. Our algorithm ADT requires only that T be nonsingular. Both our algorithms for Toeplitz systems are derived from algorithms for computing entries in the Padé table for a given power series. We prove that entries in the Padé table can be computed by the Extended Euclidean Algorithm. We describe an algorithm EMGCD (Extended Middle Greatest Common Divisor) which is faster than the algorithm HGCD of Aho, Hopcroft and Ullman, although both require time O(n log²n), and we generalize EMGCD to produce PRSDC (Polynomial Remainder Sequence Divide and Conquer) which produces any iterate in the PRS, not just the middle term, in time O(n log²n). Applying PRSDC to the polynomials U₀(x) = x²ⁿ⁺¹ and U₁(x) = a₀ + a₁x + … + a_2nx²ⁿ gives algorithm AD (Anti-Diagonal), which computes any (m, p) entry along the antidiagonal m + p = 2n of the Padé table for U₁ in time O(n log²n). Our other algorithm, MD (Main-Diagonal), computes any diagonal entry (n, n) in the Padé table for a normal power series, also in time O(n log²n). MD is related to Schönhage's fast continued fraction algorithm. A Toeplitz matrix T is naturally associated with U₁, and the (n, n) Padé approximation to U₁ gives the first column of T^?1. We show how a formula due to Trench can be used to compute the solution z of Tz = b in time O(n log n) from the first row and column of T^?1. Thus, the Padé table algorithms AD and MD give O(n log²n) Toeplitz algorithms ADT and MDT. Trench's formula breaks down in certain degenerate cases, but in such cases a companion formula, the discrete analog of the Christoffel-Darboux formula, is valid and may be used to compute z in time O(n log²n) via the fast computation (by algorithm AD) of at most four Padé approximants. We also apply our results to obtain new complexity bounds for the solution of banded Toeplitz systems and for BCH decoding via Berlekamp's algorithm.     

The best generalized inverse of the linear operator in normed linear space

Let X,Y be normed linear spaces, T∈L(X,Y) be a bounded linear operator from X to Y. One wants to solve the linear problem Ax=y for x (given y∈Y), as well as one can. When A is invertible, the unique solution is x=A^-1y. If this is not the case, one seeks an approximate solution of the form x=By, where B is an operator from Y to X. Such B is called a generalised inverse of A. Unfortunately, in general normed linear spaces, such an approximate solution depends nonlinearly on y. We introduce the concept of bounded quasi-linear generalised inverse T^h of T, which contains the single-valued metric generalised inverse T^M and the continuous linear projector generalised inverse T⁺. If X and Y are reflexive, we prove that the set of all bounded quasi-linear generalised inverses of T, denoted by G_H(T), is not empty In the normed linear space of all bounded homogeneous operators, the best bounded quasi-linear generalised inverse T^h of T is just the Moore-Penrose metric generalised inverse T^M. In the case, X and Y are finite dimension spaces Rⁿ and R^m, respectively, the results deduce the main result by G.R. Goldstein and J.A. Goldstein in 2000.     

A fast numerical method for block lower triangular Toeplitz with dense Toeplitz blocks system with applications to time-space fractional diffusion equations

Based on the circulant-and-skew-circulant representation of Toeplitz matrix inversion and the divide-and-conquer technique, a fast numerical method is developed for solving N-by-N block lower triangular Toeplitz with M-by-M dense Toeplitz blocks system with \(\mathcal {O}(MN\log N(\log N+\log M))\) complexity and \(\mathcal {O}(NM)\) storage. Moreover, the method is employed for solving the linear system that arises from compact finite difference scheme for time-space fractional diffusion equations with significant speedup. Numerical examples are given to show the efficiency of the proposed method.     

Applications of the theory of semi-embeddings to Banach space theory

Let X and Y be Banach spaces and T:X → Y an injective bounded linear operator. T is called a semi-embedding if T maps the closed unit ball of X to a closed subset of Y. (This concept was introduced by Lotz, Peck, and Porta, Proc. Edinburgh Math. Soc.22 (1979), 233–240.) It is proved that if X semi-embeds in Y, and X is separable, then X has the Radon-Nikodym property provided Y does. It is shown that if L¹ semi-embeds in Y, then Y fails the Schur property and contains a subspace isomorphic to l¹. As a consequence of the proof, it is shown that if X is a subspace of L¹, either L¹ embeds in X or l¹ embeds in $\text{L}{}^1\text{X}$. The simpler result that L¹ does not semi-embed in c₀ is treated separately. This result is used to deduce the classic result of Menchoff that there exists a singular probability measure on the circle with Fourier coefficients vanishing at infinity. Some generalizations of the notion of semi-embedding are given, and several complements and open questions are discussed.     

One step semi-explicit methods based on the Cayley transform for solving isospectral flows

This note deals with the numerical solution of the matrix differential system Y′ = [B(t,Y), Y], Y(0) = Y₀, t ⩾ 0, where Y₀ is a real constant symmetric matrix, B maps symmetric into skew-symmetric matrices, and [B(t,Y),Y] is the Lie bracket commutator of B(t,Y) and Y, i.e. [B(t,Y),Y] = B(t,Y)Y − YB(t,Y). The unique solution of (1) is isospectral, that is the matrix Y(t) preserves the eigenvalues of Y₀ and is symmetric for all t (see [1, 5]). Isospectral methods exploit the Flaschka formulation of (1) in which Y(t) is written as Y(t) = U(t)Y₀U^T(t), for t ⩾ 0, where U(t) is the orthogonal solution of the differential system U′ = B(t, UY₀U^T)U, U(0) = I, t ⩾ 0, (see [5]). Here a numerical procedure based on the Cayley transform is proposed and compared with known isospectral methods.     

Bifurcation for variational problems when the linearisation has no eigenvalues

For a bounded analytic function, ?, on the unit disk, $\text{D,}\text{let}\text{T}{}_{\mathrm{?}}\text{and}\text{M}{}_{\mathrm{?}}$ denote the operators of multiplication by ? on H²(?D) and L²(?D), respectively. In their 1973 paper, Deddens and Wong asked whether there is an analytic Toeplitz operator $\text{T}{}_{\mathrm{?}}$ that commutes with a nonzero compact operator, and whether every operator that commutes with an analytic Toeplitz operator has an extension that commutes with the corresponding multiplication operator on L². In the first part of this paper, we give an explicit example of an analytic Toeplitz operator T_φ that settles both of these questions. This operator commutes with a nonzero compact operator (a composition operator followed by an analytic Toeplitz operator). The only operators in the commutant of T_φ that extend to commute with M_φ are analytic Toeplitz operators. Although the commutant of T_φ contains more than just analytic Toeplitz operators, T_φ is irreducible. The remainder of the paper seeks to explain more fully the phenomena incorporated in this example by introducing a class of analytic functions, including the function φ, and giving additional conditions on functions g in the class to determine whether T_g commutes with nonzero compact operators, whether T_g is irreducible, and which operators in the commutant of T_g extend to the commutant of M_g. In particular, we find representations for operators in the commutant and second commutant of T_g.     

Separately radial and radial Toeplitz operators on the projective space and representation theory

We consider separately radial (with corresponding group T ⁿ ) and radial (with corresponding group U(n)) symbols on the projective space P ⁿ (C), as well as the associated Toeplitz operators on the weighted Bergman spaces. It is known that the C*-algebras generated by each family of such Toeplitz operators are commutative (see R. Quiroga-Barranco and A. Sanchez-Nungaray (2011)). We present a new representation theoretic proof of such commutativity. Our method is easier and more enlightening as it shows that the commutativity of the C*-algebras is a consequence of the existence of multiplicity-free representations. Furthermore, our method shows how to extend the current formulas for the spectra of the corresponding Toeplitz operators to any closed group lying between T ⁿ and U(n).     

A deteriorating cold standby repairable system with priority in use

In this paper, a cold standby repairable system consisting of two dissimilar components and one repairman is studied. In this system, it is assumed that the working time distributions and the repair time distributions of the two components are both exponential and component 1 is given priority in use. After repair, component 2 is “as good as new” while component 1 follows a geometric process repair. Under these assumptions, using the geometric process and a supplementary variable technique, some important reliability indices such as the system availability, reliability, mean time to first failure (MTTFF), rate of occurrence of failure (ROCOF) and the idle probability of the repairman are derived. A numerical example for the system reliability R(t) is given. And it is considered that a repair-replacement policy based on the working age T of component 1 under which the system is replaced when the working age of component 1 reaches T. Our problem is to determine an optimal policy T^∗ such that the long-run average cost per unit time of the system is minimized. The explicit expression for the long-run average cost per unit time of the system is evaluated, and the corresponding optimal replacement policy T^∗ can be found analytically or numerically. Another numerical example for replacement model is also given.     

Properties of generalized Toeplitz operators

An operatorX: is said to be a generalized Toeplitz operator with respect to given contractionsT ₁ andT ₂ ifX=T ₂XT₁ ^*. The purpose of this line of research, started by Douglas, Sz.-Nagy and Foia, and Pták and Vrbová, is to study which properties of classical Toeplitz operators depend on their characteristic relation. Following this spirit, we give appropriate extensions of a number of results about Toeplitz operators. Namely, Wintner's theorem of invertibility of analytic Toeplitz operators, Widom and Devinatz's invertibility criteria for Toeplitz operators with unitary symbols, Hartman and Wintner's theorem about Toeplitz operator having a Fredholm symbol, Hartman and Wintner's estimate of the norm of a compactly perturbed Toeplitz operator, and the non-existence of compact classical Toeplitz operators due to Brown and Halmos.Dedicated to our friend Cora Sadosky on the occasion of her sixtieth birthday     

Subnormal Toeplitz operators and the kernels of their self-commutators

In this note we give a connection between subnormal Toeplitz operators and the kernels of their self-commutators. This is closely related to P.R. Halmos's Problem 5: Is every subnormal Toeplitz operator either normal or analytic? Our main theorem is as follows: If φ∈L^∞ is such that φ and are of bounded type (that is, they are quotients of two analytic functions on the open unit disk) and if the kernel of the self-commutator of Tφ is invariant for Tφ then Tφ is either normal or analytic.     

The regular part of a semigroup of transformations with restricted range

Let T(X) be the full transformation semigroup on the set X and let T(X,Y) be the semigroup consisting of all total transformations from X into a fixed subset Y of X. It is known that $$F(X, Y)=\{\alpha\in T(X, Y): X\alpha\subseteq Y\alpha\},$$ is the largest regular subsemigroup of T(X,Y) and determines Green??s relations on T(X,Y). In this paper, we show that F(X,Y)?T(Z) if and only if X=Y and |Y|=|Z|; or |Y|=1=|Z|, and prove that every regular semigroup S can be embedded in F(S ¹,S). Then we describe Green??s relations and ideals of F(X,Y) and apply these results to get all of its maximal regular subsemigroups when Y is a nonempty finite subset of X.     

Toeplitz matrices with banded inverses

Given a Toeplitz matrix T with banded inverse [i.e., (T^?1)_ij=0 for j?i>p], we show that the elements of T can be expressed in terms of the roots of a polynomial. Then, using properties we have previously established, we generalize this result appropriately to allow singular T and show that the converse also holds. Finally, we give a sufficient condition for the decay of the elements of T as one moves away from the diagonal.