Abstract: Introduction 1. Hypercyclic and supercyclic operators 2. Hypercyclicity everywhere 3. Connectedness and hypercyclicity 4. Weakly mixing operators 5. Ergodic theory and linear dynamics 6. Beyond hypercyclicity 7. Common hypercyclic vectors 8. Hypercyclic subspaces 9. Supercyclicity and the angle criterion 10. Linear dynamics and the weak topology 11. Universality of the Riemann zeta function 12. About 'the' Read operator Appendices Notations Index Bibliography.

Abstract: A weighted theory for multilinear fractional integral operators and maximal functions is presented. Sufficient conditions for the two weight inequalities of these operators are found, including “power and logarithmic bumps” and anA ∞ condition. For one weight inequalities a necessary and sufficient condition is then obtained as a consequence of the two weight inequalities. As an application, Poincare and Sobolev inequalities adapted to the multilinear setting are presented.

Abstract: This paper concerns unitary invariants for n-tuples T:=(Tl,,Tn) of (not necessarily commuting) bounded linear operators on Hilbert spaces The author introduces a notion of joint numerical radius and works out its basic properties Multivariable versions of Berger's dilation theorem, Berger - Kato - Stampfli mapping theorem, and Schwarz's lemma from complex analysis are obtained The author studies the joint (spatial) numerical range of T in connection with several unitary invariants for n-tuples of operators such as: right joint spectrum, joint numerical radius, euclidean operator radius, and joint spectral radius He also proves an analogue of Toeplitz-Hausdorff theorem on the convexity of the spatial numerical range of an operator on a Hilbert space, for the joint numerical range of operators in the noncommutative analytic Toeplitz algebra Fn

Abstract: The notion of Lipschitz p-summing operator is introduced. A nonlinear Pietsch factorization theorem is proved for such operators, and it is shown that a Lipschitz p-summing operator that is linear is a p-summing operator in the usual sense.

Abstract: We study non-elliptic quadratic differential operators. Quadratic differential operators are non-selfadjoint operators defined in the Weyl quantization by complex-valued quadratic symbols. When the real part of their Weyl symbols is a non-positive quadratic form, we point out the existence of a particular linear subspace in the phase space intrinsically associated to their Weyl symbols, called a singular space, such that when the singular space has a symplectic structure, the associated heat semigroup is smoothing in every direction of its symplectic orthogonal space. When the Weyl symbol of such an operator is elliptic on the singular space, this space is always symplectic and we prove that the spectrum of the operator is discrete and can be described as in the case of global ellipticity. We also describe the large time behavior of contraction semigroups generated by these operators.

Abstract: In the present paper we introduce a q-analogue of the Baskakov-Kantorovich operators and investigate their weighted statistical approximation properties. By using a weighted modulus of smoothness, we give some direct estimations for error in case 0 < q < 1.

Abstract: In this paper we study the n-point correlation functions of two different families of local gauge invariant operators in = 4 supersymmetric Yang-Mills theory. The main idea is to consider the correlation functions of operators which all share a number of supersymmetries irrespective of their relative locations. We achieve this by equipping the operators with explicit space-time dependence. We provide evidence by different methods that these n-point correlators do not receive quantum corrections in perturbation theory and are hence given exactly by their tree-level result. The arguments rely on explicit checks for general four-point correlators, some five-point and six-point correlators and a more abstract calculation based on a novel topological twisting of = 4 supersymmetric Yang-Mills theory.

Abstract: We develop a natural generalization of vector-valued frame theory, we term operator-valued frame theory, using operator-algebraic methods. This extends work of the second author and D. Han which can be viewed as the mul- tiplicity one case and extends to higher multiplicity their dilation approach. We prove several results for operator-valued frames concerning duality, dis- jointeness, complementarity , and composition of operator valued frames and the relationship between the two types of similarity (left and right) of such frames. A key technical tool is the parametrization of Parseval operator val- ued frames in terms of a class of partial isometries in the Hilbert space of the analysis operator. We apply these notions to an analysis of multiframe gener- ators for the action of a discrete group G on a Hilbert space. One of the main results of the Han-Larson work was the parametrization of the Parseval frame generators in terms of the unitary operators in the von Neumann algebra gen- erated by the group representation, and the resulting norm path-connectedness of the set of frame generators due to the connectedness of the group of unitary operators of an arbitrary von Neumann algebra. In this paper we general- ize this multiplicity one result to operator-valued frames. However, both the parameterization and the proof of norm path-connectedness turn out to be necessarily more complicated, and this is at least in part the rationale for this paper. Our parameterization involves a class of partial isometries of a different von Neumann algebra. These partial isometries are not path-connected in the norm topology, but only in the strong operator topology. We prove that the set of operator frame generators is norm pathwise-connected precisely when the von Neumann algebra generated by the right representation of the group has no minimal projections. As in the multiplicity one theory there are analogous results for general (non-Parseval) frames.

Abstract: Sufficient conditions for the boundedness of the Hausdorff operators in the Hardy spaces H, 0 < p < 1, on the real line are proved. Two related negative results are also given.

Abstract: Some power inequalities for the numerical radius of a prod- uct of two operators in Hilbert spaces with applications for commutators and self-commutators are given.

Abstract: We give a complete picture of the boundedness and compactness of the products of integral-type operators and composition operators from the mixed norm space to Bloch-type spaces of holomorphic functions on the unit disk.

Abstract: We prove quantitative bounds on the eigenvalues of non-selfadjoint unbounded operators obtained from selfadjoint operators by a perturbation that is relatively-Schatten. These bounds are applied to obtain new results on the distribution of eigenvalues of Schroedinger operators with complex potentials.

Abstract: We present an orthogonal basis of gauge invariant operators constructed from some complex matrices for the free matrix field, where operators are expressed with the help of Brauer algebra. This is a generalisation of our previous work for a signle complex matrix. We also discuss the matrix quantum mechanics relevant to = 4 SYM on S3 × R. A commuting set of conserved operators whose eigenstates are given by the orthogonal basis is shown by using enhanced symmetries at zero coupling.

Abstract: We study linear differential operators with unbounded operator-valued coefficients acting on homogeneous spaces of functions on the semi-axis. We obtain necessary and sufficient conditions for such operators to be invertible or Fredholm. Essential use is made of the spectral theory of difference relations and semigroups of linear relations (multi-valued linear operators).

Abstract: We prove that the product of finitely many Toeplitz operators oil the Hardy space is zero if and only if at least one of the operators is zero. We use some new vector-valued techniques that not only lead to a vector-valued version of this result but also appear to be needed in the scalar case.

Abstract: It is known that if a rearrangement invariant function space E on (0,1) has an unconditional basis then each linear continuous operator on E is a sum of two narrow operators. On the other hand, the sum of two narrow operators in L1 is narrow. To find a general approach to these results, we extend the notion of a narrow operator to the case when the domain space is a vector lattice. Our main result asserts that the set Nr(E, F )o f all nar- row regular operators is a band in the vector lattice Lr(E, F) of all regular operators from a non-atomic order continuous Banach lattice E to an order continuous Banach lattice F. The band generated by the disjointness preserv- ing operators is the orthogonal complement to Nr(E, F )i nLr(E, F). As a consequence we obtain the following generalization of the Kalton-Rosenthal theorem: every regular operator T : E → F from a non-atomic Banach lattice E to an order continuous Banach lattice F has a unique representation as T = TD + TN where TD is a sum of an order absolutely summable family of disjointness preserving operators and TN is narrow. Mathematics Subject Classification (2000). Primary 47B65; secondary 47B38.

Abstract: This review covers an important domain of p-adic mathematical physics — quantum mechanics with p-adic valued wave functions. We start with basic mathematical constructions of this quantum model: Hilbert spaces over quadratic extensions of the field of p-adic numbers ℚ p , operators — symmetric, unitary, isometric, one-parameter groups of unitary isometric operators, the p-adic version of Schrodinger’s quantization, representation of canonical commutation relations in Heisenberg andWeyl forms, spectral properties of the operator of p-adic coordinate.We also present postulates of p-adic valued quantization. Here observables as well as probabilities take values in ℚ p . A physical interpretation of p-adic quantities is provided through approximation by rational numbers.

Abstract: We introduce sequences of operators Vn of King’s type and study them in regard to uniform convergence, global smoothness preservation, the approximation of decreasing and convex functions, the validity of a Voronovskaja- type theorem and a recursion formula generalizing a corresponding result for the classical Bernstein operators.

Abstract: We prove that for compatible weakly nonlocal Hamiltonian and symplectic operators, hierarchies of infinitely many commuting local symmetries and conservation laws can be generated under some easily verified conditions no matter whether the generating Nijenhuis operators are weakly nonlocal or not. We construct a recursion operator of the two-dimensional periodic Volterra chain from its Lax representation and prove that it is a Nijenhuis operator. Furthermore we show that this system is a (generalized) bi-Hamiltonian system. Rather surprisingly, the product of its weakly nonlocal Hamiltonian and symplectic operators gives rise to the square of the recursion operator.

Abstract: In this paper we generalize the Sato theory to the extended bigraded Toda hierarchy (EBTH). We revise the definition of the Lax equations,give the Sato equations, wave operators, Hirota bilinear identities (HBI) and show the existence of $tau$ function $\tau(t)$. Meanwhile we prove the validity of its Fay-like identities and Hirota bilinear equations (HBEs) in terms of vertex operators whose coefficients take values in the algebra of differential operators. In contrast with HBEs of the usual integrable system, the current HBEs are equations of product of operators involving $e^{\partial_x}$ and $\tau(t)$.

Abstract: We discuss some results for q-analogues of Bernstein and Stancu operators. Moreover, a q-analogue of an operator of A. Lupas is introduced and investigated.

Abstract: Very recently Gupta (Appl. Math. Comput. 197(1), 172–178, 2008) introduced the q-Durrmeyer operators Dn,qf and studied some approximation properties of such operators. In the present paper, we extend the studies and here we obtain some local and global direct results for the q-Durrmeyer type operators. Furthermore, we establish a simultaneous approximation theorem for Dn,qf, where f is a polynomial.