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This paper gives necessary and sufficient conditions for an essentially normal operator T to have its C ⁎ -algebra C ⁎ ( T ) generated by a complex symmetric operator. Also it is completely determined when C ⁎ ( T ) is ⁎-isomorphic to a C ⁎ -algebra singly generated by complex symmetric operators. These both depend only on the singular part of T.

We demonstrate new abstract characterizations for unital and non-unital operator spaces. We characterize unital operator spaces in terms of the cone of accretive operators (operators whose real part is positive). Defining the gauge of an operator $T \in B(H)$ to be $\|Re(T)_+\|$, we demonstrate an abstract characterization of operator spaces up to complete gauge-isometry. Both of these characterizations preserve the structure of the self-adjoint, positive, and accretive operators, as well as the operator norm. We show that an operator space with a given matrix ordering of positive or accretive cones can be represented completely isometrically and completely order isomorphically if and only if each positive cone is normal, in the sense that $x \leq y \leq z$ implies that $\|y\| \leq \max(\|x\|,\|z\|)$ at each matrix level. This is achieved by showing that normal matrix ordered operator spaces are induced by gauges. We show that inducing gauges are not unique in general. Finally, we show that completely positive completely contractive linear maps on non-unital operator spaces extend to any containing operator system if and only if the operator space is induced by a unique gauge., Comment: 18 pages. Exposition significantly updated thanks to helpful comments from the editors and referees. To appear in the Journal of Mathematical Analysis and Applications

In this paper a twofold inverse problem for orthogonal matrix functions in the Wiener class is considered. The scalar-valued version of this problem was solved by Ellis and Gohberg in 1992. Under reasonable conditions, the problem is reduced to an invertibility condition on an operator that is defined using the Hankel and Toeplitz operators associated to the Wiener class functions that comprise the data set of the inverse problem. It is also shown that in this case the solution is unique. Special attention is given to the case that the Hankel operator of the solution is a strict contraction and the case where the functions are matrix polynomials., Comment: 15 pages

We prove the higher integrability and higher order imbedding theorems of the composition of homotopy operator and Green's operator. We also study analogues of the homotopy operator and Green's operator, respectively.

We consider a family of the discrete Schrodinger operators H λ μ , depending on two parameters, in the d-dimensional lattice with a potential constructed via the delta function and the shift operator. The existence of eigenvalues, threshold eigenvalues and threshold resonances and their dependence on the parameters of the operator and dimension of the lattice are explicitly derived.

We present an operator equation characterization of a dual truncated Toeplitz (DTT) operator using the compressed shift operator. This characterization is an analogue of the classical characterization of a Toeplitz operator [2] and the more recent characterization of a truncated Toeplitz operator [32] . As applications of this characterization, we give short proofs and refinements of some basic properties of DTT operators which were obtained in [9] , [10] by more complicated methods. Furthermore, we determine when a DTT operator is normal, and when two DTT operators commute. We identify the commutant of the compressed shift which is a commutant lifting theorem in spirit.

The purpose of this note is to describe the similarity of Cowen–Douglas operators with index one by considering the difference of the curvatures of the corresponding vector bundles. We define a subclass of operators of B 1 ( D ) that includes the weighted shift operators with weighted sequence { [ ( k + 1 ) ( k + α ) ] 1 / 2 } k = 0 ∞ , α ≥ 1 and consider when an operator in B 1 ( D ) is similar to some operator in this operator class.

In this paper we investigate a new class of operators called weighted shifts on directed trees introduced recently in [Z. J. Jablonski, I. B. Jung and J. Stochel, A Non-hyponormal Operator Generating Stieltjes Moment Sequences, J. Funct. Anal. 262 (2012), no. 9, 3946--3980.]. This class is a natural generalization of the so called weighted bilateral, unilateral and backward shift operators. In the first part of the paper we calculate the asymptotic limit and the isometric asymptote of a contractive weighted shift on a directed tree and that of the adjoint. Then we use the asymptotic behaviour and similarity properties to deal with cyclicity. We also show that a weighted backward shift operator is cyclic if and only if there is at most one zero weight., Comment: 22 pages

We introduce a generalization of rectangular Herz spaces defined in [3] and prove boundedness of a version of the n-dimensional Hardy operator on them. We also study the action of the Hausdorff operator on some subspaces of BMO.

In this paper, we give a characterization of a complex symmetric Toeplitz operator T φ on the Hardy space H 2 . Moreover, if T φ is a complex symmetric Toeplitz operator, we provide a necessary and sufficient condition for T φ to be normal. Finally, we investigate these T φ with finite symbols.

The subnormal completion problem for d-variable weighted shifts is considered. Necessary and sufficient conditions are obtained for a collection of initial weights C = { ( α γ ( 1 ) , … , α γ ( d ) ) } γ ∈ Γ , where Γ is from a family of finite indexing sets which includes { γ ∈ N 0 d : 0 ≤ | γ | ≤ m } , to give rise to a d-variable subnormal weighted shift operator whose initial weights are given by C . The conditions are communicated in terms of a new solution of a corresponding truncated K-moment problem. The case when d = 2 and all cubic moments are known is investigated in detail. In particular, using the solution of the subnormal completion problem in d-variables presented here, an easily checked concrete sufficient condition is given for a solution to the cubic subnormal completion problem in 2-variables and also an example of a collection of weights in 2-variables with cubic moments is provided which satisfies natural positivity conditions yet does not admit a subnormal completion.

As it is well-known, for any operator T on a complex separable Hilbert space, T has the polar decomposition T = U | T | , where U is a partial isometry and | T | is the non-negative operator ( T ⁎ T ) 1 2 . In this paper, we will give a decomposition theorem in a new sense that | T | will be replaced by a strongly irreducible operator. More precisely, for any operator T and any e > 0 , there exists a decomposition T = ( U + K ) S , where U is a partial isometry, K is a compact operator with | | K | | e and S is strongly irreducible.

A complex scalar λ is said to be an extended eigenvalue of a bounded linear operator T on a complex Banach space if there is a non-zero operator X such that T X = λ X T . Such an operator X is called an extended eigenoperator of T corresponding to the extended eigenvalue λ . The purpose of this paper is to give a description of the extended eigenvalues for the discrete Cesaro operator C 0 , the finite continuous Cesaro operator C 1 and the infinite continuous Cesaro operator C ∞ defined on the complex Banach spaces l p , L p [ 0 , 1 ] and L p [ 0 , ∞ ) for 1 p ∞ by the expressions ( C 0 f ) ( n ) : = 1 n + 1 ∑ k = 0 n f ( k ) , ( C 1 f ) ( x ) : = 1 x ∫ 0 x f ( t ) d t , ( C ∞ f ) ( x ) : = 1 x ∫ 0 x f ( t ) d t . It is shown that the set of extended eigenvalues for C 0 is the interval [ 1 , ∞ ) , for C 1 it is the interval ( 0 , 1 ] , and for C ∞ it reduces to the singleton {1}.

We characterize the approximation of functions in uniform norm by classical Meyer-Konig and Zeller operator. Using the closed connection between the approximation by Meyer-Konig and Zeller operator and the weighted approximation by Baskakov operator, we prove a strong converse inequality of type A in terms of the K -functional.

The lattice of closed invariant subspaces of the Volterra operator acting on $L^2(0,1)$ was completely described by Sarason. On the other hand, he explicitly found the lattice of closed invariant subspaces of the shift plus Volterra operator on $L^2(0,1)$. Inspired by Sarason's results, we find the lattice of closed invariant subspaces of the shift plus complex Volterra operator acting on the Hardy space, Comment: The article has been accepted for publication in the Journal of Mathematical Analysis and Applications

The stabilities of fusion frames under operator perturbation (simple named operator perturbation of the fusion frames), is the study object of this paper. We provide a way as follows for studying the operator perturbation of fusion frames. Firstly, we consider the operator perturbation of fusion frame sequences by the gap between two closed subspaces and discuss the relationships between the operator perturbation of fusion frames and the operator perturbation of fusion frame sequences. Lastly, we transform the operator perturbation of fusion frames into the operator perturbation of fusion frame sequences based on these relationships. Our results obtained this way generalize the remarkable results which have been obtained by Casazza, Kutyniok, Asgari, Gavruta and Zhu.

Properties of m-selfadjoint and m-isometric operators have been investigated by several researchers. Particularly interesting to us are algebraic properties of nilpotent perturbations of such operators. McCullough and Rodman showed in the nineties that if Q n = 0 and A is a selfadjoint operator commuting with Q then the sum A + Q is a ( 2 n − 1 ) -selfadjoint operator. Very recently, Bermudez, Martinon, and Noda proved a similar result for nilpotent perturbations of isometries. Via a new approach, we obtain simple proofs of these results and other generalizations to operator roots of polynomials.

In this paper, we use the q -Chu–Vandermonde formula to prove two new operator identities, which are the extensions of Liu's results. These two q -exponential operator identities are used to derive some q -summation formulas and q -integrals.

The Kerzman–Stein operator is the skew-hermitian part of the Cauchy operator defined with respect to an unweighted hermitian inner product on the boundary. For bounded regions with smooth boundary, the Kerzman–Stein operator is compact on the Hilbert space of square integrable functions. Here we give an explicit computation of its Hilbert–Schmidt norm for a family of simply connected regions. We also give an explicit computation of the Cauchy operator acting on an orthonormal basis, and we give estimates for the norms of the Kerzman–Stein and Cauchy operators on these regions. The regions are the first regions that display no apparent Mobius symmetry for which there now is explicit spectral information.

We prove that if an isometry A and a nilpotent operator Q of order n commute, then A + Q is a strict ( 2 n − 1 ) -isometry. As an application of the main result, we prove that A + Q cannot be N -supercyclic for any N . Also, we find an m -isometry with prescribed spectrum K , where K is the closed unit disk or a compact subset of the unit circle.

Characterization of invariant subspaces with respect to shift operator in the class of uniformly convex rearrangement invariant Kothe function spaces is given.

We reveal a connection between operator monotone functions and orthogonal polynomials. Especially, we express an operator monotone function with a Jacobi operator, and show that it is a limit of rational operator monotone functions. Further we prove that the ‘principal inverse’ of an orthogonal polynomial is operator monotone and hence it has a holomorphic extension to the open upper half plane, namely a Pick function (or Nevanlinna function).

This paper is devoted to the study of the domains of singularities in C (resp. in C × C ) of the operator valued functions k ↦ V k and ( k , k ′ ) ↦ V k , k ′ , where V k and V k , k ′ are the operators which intertwine the derivative operator d d x with respectively the Dunkl operator T ( k ) f ( x ) = d f d x ( x ) + k f ( x ) − f ( − x ) x and the Jacobi–Cherednik operator T ( k , k ′ ) f ( x ) = f ′ ( x ) + ( k coth ( x ) + k ′ tanh ( x ) ) ( f ( x ) − f ( − x ) ) − ( k + k ′ ) f ( − x ) . We also determine the singularities of the inverses and the duals of these operators V k and V k , k ′ by analytic methods and we show that some of them are entire functions whereas others are only meromorphic functions on C and C × C respectively.

The best polynomial approximation operator was recently extended by one of the authors from L p to L p − 1 . In this paper, we study weak and strong inequalities for maximal operators related with the extended best polynomial approximation operator. As an application, we obtain norm convergence of the coefficients of the best polynomial approximation.

When A ∈ B ( H ) and B ∈ B ( K ) are given, we denote by M C the operator acting on the infinite dimensional separable Hilbert space H ⊕ K of the form M C = ( A C 0 B ) . In this paper, we first give some necessary and sufficient conditions for M C to be a left invertible operator (an upper semi-Weyl, upper semi-Fredholm) operator for some C ∈ B ( K , H ) , which extend the corresponding results in Cao et al. (2006) [4] , Cao and Meng (2005) [5] , Hwang and Lee (2001) [12] and Li and Du (2006) [15] . Then we present some counter-examples.

In this paper, we show that every ( p , k ) -quasihyponormal operator has a scalar extension and give some spectral properties of the scalar extensions of ( p , k ) -quasihyponormal operators. As a corollary, we get that such an operator with rich spectrum has a nontrivial invariant subspace. Finally, we prove that the sum of a p-hyponormal operator and an algebraic operator which are commuting is subscalar.

Let A and B be positive operators on a Banach lattice E such that the commutator C = A B − B A is also positive. The paper continues the investigation of the spectral properties of C initiated in J. Bracic et al. (in press) [3] . If the sum A + B is a Riesz operator and the commutator C is a power compact operator, then C is a quasi-nilpotent operator having a triangularizing chain of closed ideals of E . If we assume that the operator A is compact and the commutator A C − C A is positive, the operator C is quasi-nilpotent as well. We also show that the commutator C is not invertible provided the resolvent set of C is connected.

We show that the Libera operator, L, on some spaces of analytic functions is a continuous extension of the conjugate of the Cesaro operator. Results on L acting on various spaces are obtained. In particular, L maps the Bloch space into BMOA. We also prove some results on the best approximation by polynomials in Hardy and Bergman spaces.

This paper considers existence of multiple positive fixed points for some nonlinear operators, a particular case of the operators is sum of an e-concave operator and an e-convex operator. Then we apply the results to nonlinear integral equations.

We consider the generalized shift operator, associated with the Laplace–Bessel differential operator Δ B = ∑ i = 1 n ∂ 2 ∂ x i 2 + ∑ i = 1 k γ i x i ∂ ∂ x i . The maximal operator M γ ( B -maximal operator) and the Riesz potential I γ α ( B -Riesz potential), associated with the generalized shift operator are investigated. At first, we prove that the B -maximal operator M γ is bounded from the B -Morrey space L p , λ , γ to L p , λ , γ for all 1 p ∞ and 0 ⩽ λ n + | γ | . We prove that the B -Riesz potential I γ α , 0 α n + | γ | is bounded from the B -Morrey space L p , λ , γ to L q , λ , γ if and only if α / ( n + | γ | − λ ) = 1 / p − 1 / q , 1 p ( n + | γ | − λ ) / α . Also we prove that the B -Riesz potential I γ α is bounded from the B -Morrey space L 1 , λ , γ to the weak B -Morrey space WL q , λ , γ if and only if α / ( n + | γ | − λ ) = 1 − 1 / q .

We formulate the Taylor series expansion for the q-numerical radius of a weighted shift operator with periodic weights near q = 0 . Coefficients up to the fourth order in the expansion are found via the perturbation theory of Hermitian matrices.

In this paper, we will introduce the generalized operator equilibrium problem and generalized operator quasi-equilibrium problem which generalize the operator equilibrium problem due to Kazmi and Raouf [K.R. Kazmi, A. Raouf, A class of operator equilibrium problems, J. Math. Anal. Appl. 308 (2005) 554–564] into multi-valued and quasi-equilibrium problems. Using a Fan–Browder type fixed point theorem in [S. Park, Foundations of the KKM theory via coincidences of composites of upper semicontinuous maps, J. Korean Math. Soc. 31 (1994) 493–519] and an existence theorem of equilibrium for 1-person game in [X.-P. Ding, W.K. Kim, K.-K. Tan, Equilibria of non-compact generalized games with L ∗ -majorized preferences, J. Math. Anal. Appl. 164 (1992) 508–517] as basic tools, we prove new existence theorems on generalized operator equilibrium problem and generalized operator quasi-equilibrium problem which includes operator equilibrium problems.

This paper concerns the Cesaro operator acting on various spaces of analytic functions on the unit disc. The remarkable fact that this operator is subnormal when acting on the Hardy space H2 has lead to extensive studies of its spectral picture on other spaces of this type. We present some of the methods that have been used to obtain information about the spectrum of the Cesaro operator acting on Hardy and Bergman spaces and give a unified approach to these problems which also yields new results in this direction. In particular, we prove that the Cesaro operator is subdecomposable on H1 and on the standard weighted Bergman spaces , α0.

The point source of this work is Seleznev's theorem which asserts the existence of a power series which satisfies universal approximation properties in C ∗ . The paper deals with a strengthened version of this result. We establish a double approximation theorem on formal power series using a weighted backward shift operator. Moreover we give strong conditions that guarantee the existence of common universal series of an uncountable family of weighted backward shift with respect to the simultaneous approximation. Finally we obtain results on admissible growth of universal formal power series. We especially prove that you cannot control the defect of analyticity of such a series even if there exist universal series in the well-known intersection of formal Gevrey classes.

In this paper, we introduce the generalized shift operator generated by the Gegenbauer differential operator D λ = ( x 2 − 1 ) 1 2 − λ d d x ( x 2 − 1 ) λ + 1 2 d d x , and define a generalized convolution ⊗ on the half-line corresponding to the Gegenbauer differential operator. We investigate the Calderon reproducing formula associated with the convolution ⊗ involving finite Borel measures, leading to results on the L p -norm and pointwise approximation for functions on the half-line.

This paper is focused on the operator inequalities of the Bohr type. We will give a new and transparent proof for the operator Bohr inequality through an absolute value operator identity, show some related operator inequalities by means of 2 × 2 (block) operator matrices, and finally we will present a generalization of the operator Bohr inequality for multiple operators.

We study the degenerate elliptic differential operator of the second order in the divergence form. The operator is assumed to be symmetric. The weight function which is describing the degeneration of the coefficients (or singularity) assumed to be in the Muckenhoupt class. We prove the uniform estimates for the fundamental solution of this operator and obtain the conditions which guarantee the absolute and uniform convergence of Fourier series in eigenfunctions. These results might be applied to the ground of Fourier method.

In this paper, we study mixed variational-like inclusions and J η -proximal operator equations in Banach spaces. It is established that mixed variational-like inclusions in real Banach spaces are equivalent to fixed point problems. We also establish a relationship between mixed variational-like inclusions and J η -proximal operator equations. This equivalence is used to suggest an iterative algorithm for solving J η -proximal operator equations.

In this paper, we study the existence and multiplicity of solutions of the operator equation K f u = u in the real Hilbert space L 2 ( G ) . Under certain conditions on the linear operator K , we establish the conditions on f which are able to guarantee that the operator equation has at least one solution, a unique solution, and infinitely many solutions, respectively. The monotone operator principle and the critical point theory are employed to discuss this problem, respectively. In argument, quadratic root operator K 1 / 2 and its properties play an important role. As an application, we investigate the existence and multiplicity of solutions to fourth-order boundary value problems for ordinary differential equations with two parameters, and give some new existence results of solutions.

This paper characterizes sequences of vectors that are the orbits of a linear operator and sequences of vectors in a Hilbert space that are orbits of a unitary operator. The latter is applied to time series. Sequences of vectors in a Hilbert space that generalize random walks are also shown to be the orbits of a bounded linear operator.

This paper is intended to discuss the problems of solving system of linear operator equations and a kind of nonlinear operator equation A u B u + C u = f . For system of linear operator equations, we obtain a criterion about the existence of solutions. If it has solutions, we get the analytic representation of its minimum norm solution and the structure of its solution space; for nonlinear operator equation A u B u + C u = f , with the help of the discussion on how to solve the system of linear operator equations, the exact solution is given. Besides, we obtain the representation of all its solutions.

We show that any countable family of operators of the form P ( B ) , where P is a non-constant polynomial and B is the backward shift operator on ω, the countably infinite product of lines, has a common hypercyclic subspace.

This paper is concerned with an operator equation on half-line. Some theorems for the existence of its positive solutions are obtained by using the Krasnosel'skii–Guo theorem on cone expansion and compression in a special function space.

This paper deals with the unilateral backward shift operator T on a Bargmann space F ( C ) . This space can be identified with the sequence space l 2 ( N ) . We use the hypercyclicity criterion of Bes, Chan, and Seubert and the program of K.-G. Grosse-Erdmann to give a necessary and sufficient condition in order that T be a chaotic operator. The chaoticity of differentiation which correspond to the annihilation operator in quantum radiation field theory is in view, since the Bargmann space is an infinite-dimensional separable complex Hilbert space.

The relation between power boundedness and similarity to a contraction has been thoroughly investigated by Cassier (1999) for a certain class of operators on Hilbert space. We explore the application of those results to one-dimensional perturbations of the shift operator. The criteria obtained lead to various connections with function theory problems.

We consider a maximal operator defined on Rn which is related to the Cesàro α continuity of functions. We characterize the weights w for which the maximal operator is of weak type, strong type and restricted weak type (p,p) with respect to the measure w(x)dx.

The main object of this paper is to establish a lower bound of the entropy numbers for powers of weighted shift operators on lp spaces, 1 ≤ p ≤ ∞, and we obtain some results of Pietsch. Furthermore, we get an interesting inequality between entropy numbers and s numbers of these shifts and, as a simple application of this inequality, we have, for q ∈ (0, ∞), that the two operator ideals E(e)q(lp, lp) and S(s)q(lp, lp) are identical.

An Lpμ-boundedness result for the pseudo-differential operator associated with the Bessel operator is obtained.

The paper considers several different stability notions for discrete-time linear time-varying systems specified by almost-periodic kernels. Various input-output stability notions are analysed, including those in which the signal spaces consist of bounded power and asymptotically almost-periodic functions. Further, it is proved that every stable time-varying discrete time system commuting with a polynomial in the shift operator is periodic. An example shows that this result does not hold for unstable time-varying systems.

A causal input–output system operating for all time from the indefinite past to the indefinite future may be described by a function space for inputs, a function space for outputs, and a causal operator mapping the input space into the output space. The state of such a system at any instant is defined here as an operator from the space of possible future inputs to that of future outputs. This operator is called the natural state. The output space is taken to be a time-shift-invariant normed linear function space, and the input space is either also such a space or a time-shift-invariant subset thereof. There is flexibility allowed in the choice of these spaces. Both the input–output operator and the operator giving the natural state are themselves taken to be elements of normed linear spaces with one of a particular family of norms called N -power norms. The general development applies to nonlinear and time-varying systems. Continuity and boundedness of the natural state (as an operator) and properties of the natural state and its trajectory as related to the input–output description of the system are investigated. Two examples are presented.