On Operators Satisfying T＊｜T2｜T≥ T＊｜T＊｜2T

Let T be a bounded linear operator on a complex Hilbert space H. In this paper we introduce a new class denoted by l-＊-A, of operators satisfying T＊｜T2｜T≥ T＊｜T＊｜2T, and we prove the basic properties of these operators. Using these results, we also prove that if T or T＊ ∈l-＊-A, then w（f（T）） = f（w（T））, σea（f（T）） = f（σea（T）） for every f C H（σ（T））, where g（σ（T）） denotes the set of all analytic functions on an open neighborhood of σ（T）.     

Xia Spectrum for Some Class of Operators

In this paper, we introduce Xia spectra of n-tuples of operators satisfying |T ²| ≥ U|T ²|U* for the polar decomposition of T = U|T| and we extend Putnam’s inequality to these tuples [7]. This research is partially supported by Grant-in-Aid Research No.17540176.     

Nonexistence of invariant graphs in all supercritical energy levels of mechanical Lagrangians in T 2

Let (T², g) be a smooth Riemannian structure in the torus T². We show that given ε > 0 and any C^∞ function U : T² → ℝ there exists a C¹ function U_ε with Lipschitz derivatives that is ε-C⁰ close to U for which there are no continuous invariant graphs in any supercritical energy level of the mechanical Lagrangian L_ε : TT² → ℝ given by . We also show that given n ∈ ℕ, the set of C^∞ potentials U : T² → ℝ for which there are no continuous invariant graphs in any supercritical energy level E ≤ n of is C⁰ dense in the set of C^∞ functions. Partially supported by CNPq, FAPERJ-Cientistas do nosso estado.     

Approximating Solutions of Maximal Monotone Operators in Hilbert Spaces

Let H be a real Hilbert space and let T: H→2^H be a maximal monotone operator. In this paper, we first introduce two algorithms of approximating solutions of maximal monotone operators. One of them is to generate a strongly convergent sequence with limit vT⁻¹0. The other is to discuss the weak convergence of the proximal point algorithm. Next, using these results, we consider the problem of finding a minimizer of a convex function. Our methods are motivated by Halpern's iteration and Mann's iteration.     

Ergodic properties of operators in some semi-Hilbertian spaces

This article deals with linear operators T on a complex Hilbert space ?, which are bounded with respect to the seminorm induced by a positive operator A on ?. The A-adjoint and A ^1/2-adjoint of T are considered to obtain some ergodic conditions for T with respect to A. These operators are also employed to investigate the class of orthogonally mean ergodic operators as well as that of A-power bounded operators. Some classes of orthogonally mean ergodic or A-ergodic operators, which come from the theory of generalized Toeplitz operators are considered. In particular, we give an example of an A-ergodic operator (with an injective A) which is not Cesàro ergodic, such that T ^?* is not a quasiaffine transform of an orthogonally mean ergodic operator.     

On Positive Operators on Some Ordered Banach Spaces

In this paper we study to what extent some classical results concerning operators T, from a -space to a Banach space, or from a Banach space to a L ¹-space, can be precised, when the Banach spaces involved are ordered (by a normal cone in the first case, by a closed generating proper convex cone in the second case) and when the operators T are positive.     

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     

A variational theory for monotone vector fields

Monotone vector fields were introduced almost 40 years ago as nonlinear extensions of positive definite linear operators, but also as natural extensions of gradients of convex potentials. These vector fields are not always derived from potentials in the classical sense, and as such they are not always amenable to the standard methods of the calculus of variations. We describe here how the selfdual variational calculus, developed recently by the author, provides a variational approach to PDEs and evolution equations driven by maximal monotone operators. To any such vector field T on a reflexive Banach space X, one can associate a convex selfdual Lagrangian L _T on the phase space X × X ^* that can be seen as a “potential” for T, in the sense that the problem of inverting T reduces to minimizing a convex energy functional derived from L _T . This variational approach to maximal monotone operators allows their theory to be analyzed with the full range of methods—computational or not—that are available for variational settings. Standard convex analysis (on phase space) can then be used to establish many old and new results concerned with the identification, superposition, and resolution of such vector fields. Dedicated to Felix Browder on his 80th birthday     

Continuous homogeneous selections of set-valued metric generalized inverses of linear operators in Banach spaces

In this paper,continuous homogeneous selections for the set-valued metric generalized inverses T of linear operators T in Banach spaces are investigated by means of the methods of geometry of Banach spaces.Necessary and sufficient conditions for bounded linear operators T to have continuous homogeneous selections for the set-valued metric generalized inverses T are given.The results are an answer to the problem posed by Nashed and Votruba.     

Regular Couplings of Dissipative and Anti-Dissipative Unbounded Operators,Asymptotics of the Corresponding Non-Dissipative Processes and the Scattering Theory

In this paper a triangular model of a class of unbounded non-selfadjoint K ^r-operators A presented as a coupling of dissipative and anti-dissipative operators in a Hilbert space with real absolutely continuous spectra and with different domains of A and A ^* is considered. The asymptotic behaviour of the corresponding non-dissipative processes T_tf = e^itAf, generated from the semigroups T_t with generators iA, as t → ± ∞ are obtained. The strong wave operators, the scattering operator for the couple (A^*, A) and the similarity of A and the operator of multiplication by the independent variable are obtained explicitly. The considerations are based on the triangular models and characteristic functions of A. Kuzhel for unbounded operators and the limit values of the multiplicative integrals, describing the characteristic function of the considered model. Partially supported by Grant MM-1403/04 of MESC and by Scientific Research Grant 27/25.02.2005 of Shumen University.     

On Ideal Operators

Let E, F be Archimedean Riesz spaces. We consider operators that map ideals of E to ideals of F and operators T for which, T ^–1 (I) is an ideal in E, for each ideal I in F. We study the properties of such operators and investigate their relation to disjointness preserving operators.     

Abstract,weighted, and multidimensional Adamjan-Arov-Krein theorems,and the singular numbers of Sarason commutants

The classical Adamjan-Arov-Krein (A-A-K) theorem relating the singular numbers of Hankel operators to best approximations of their symbols by rational functions is given an abstract version. This provides results for Hankel operators acting in weightedH ²(T; ), as well as inH ²(T ^d ), and an A-A-K type extension of Sarason's interpolation theorem. In particular, it is shown that all compact Hankel operators inH ²(T ^d ) are zero.Author partially supported by NSF grant DMS89-11717.     

Korovkin-type theorems and applications

Let {T _n } be a sequence of linear operators on C[0,1], satisfying that {T _n (e _i )} converge in C[0,1] (not necessarily to e _i ) for i = 0,1,2, where e _i = t ⁱ . We prove Korovkin-type theorem and give quantitative results on C ²[0,1] and C[0,1] for such sequences. Furthermore, we define King’s type q-Bernstein operator and give quantitative results for the approximation properties of such operators.      

A Note on a Class of Operators

The aim of this note is to study the spectral properties of the LUECKE's class R of operators T such that ‖(T – zI)^?1‖=1/d(z, W(T)) for all z?CLW(T), where CLW(T) is the closure of the numerical range W(T) of T and d(z, W(T)) is the distance from z to W(T). The main emphasis is on the investigation of those properties of operators of class R which are either similar to or distinct from those of operators satisfying the growth condition (G₁).     

Spectra of completely log-hyponormal operators

An operatorT on a Hilbert space is called log-hyponormal if it is invertible and log(T ^* T)≥log(T ^* T). In this paper we study spectral properties of completely log-hyponormal operators. Dedicated to professor Robin Harte on his sixtieth birthday This research is partially supported by Grant-in-Aid Scientific Research (No. 09640229).     

Polar decompositions and related classes of operators in spaces ∏κ

Polar decompositions with respect to an indefinite inner product are studied for bounded linear operators acting on a space. Criteria are given for existence of various forms of the polar decompositions, under the conditions that the range of a given operatorX is closed and that zero is not an irregular critical point of the selfadjoint operatorX ^[*]X. Both real and complex spaces are considered. Relevant classes of operators having a selfadjoint (in the sense of the indefinite inner product) square root, or a selfadjoint logarithm, are characterized.The work of this author was partially supported by INdAM-GNCS and MURSTThe work of this author was partially supported by NSF grant DMS-9988579.     

Relative T -Injective Modules and Relative T -Flat Modules

Let T be a Wakamatsu tilting module. A module M is called (n, T)-copure injective (resp. (n, T)-copure flat) if ɛ _T ¹ (N, M) = 0 (resp. Γ₁ ^T (N, M) = 0) for any module N with T-injective dimension at most n (see Definition 2.2). In this paper, it is shown that M is (n, T)-copure injective if and only if M is the kernel of an I _n (T)-precover f: A → B with A ∈ Prod T. Also, some results on Prod T-syzygies are presented. For instance, it is shown that every nth Prod T-syzygy of every module, generated by T, is (n, T)-copure injective.     

Polaroid operators,SVEP and perturbed Browder,Weyl theorems

A Banach space operatorT ɛB(X) is polaroid,T ɛP, if the isolated points of the spectrum ofT are poles of the resolvent ofT. LetPS denote the class of operators inP which have have SVEP, the single-valued extension property. It is proved that ifT is polynomiallyPS andA ɛB(X) is an algebraic operator which commutes withT, thenf(T+A) satisfies Weyl’s theorem andf(T ^*+A ^*) satisfiesa-Weyl’s theorem for everyf which is holomorphic on a neighbourhood of σ(T+A).     

Non-supercyclicity of Volterra Convolution and Related Operators

We show that if V ^α (α > 0) is the Riemann-Liouville fractional integration operator and T is an invertible operator on L ²(0, 1) which commutes with V , then TV ^α is not supercyclic on L ²(0, 1); in particular, many Volterra convolution operators are not supercyclic. The technique is based on an argument used by Gallardo-Gutiérrez and Montes-Rodríguez to show that V is not supercyclic.     

Perimeter preserver of matrices over semifields

For a rank-1 matrix A = ab ^t, we define the perimeter of A as the number of nonzero entries in both a and b. We characterize the linear operators which preserve the rank and perimeter of rank-1 matrices over semifields. That is, a linear operator T preserves the rank and perimeter of rank-1 matrices over semifields if and only if it has the form T(A) = U AV, or T(A) = U A ^t V with some invertible matrices U and V. This work was supported by the research grant of the Cheju National University in 2006.