On the strict monotonicity of spectral radii for classes of bounded positive linear operators

Strict monotonicity of the spectral radii of bounded, positive, ordered linear operators is investigated. It is well-known that under reasonable assumptions, the spectral radii of two ordered positive operators enjoy a non-strict inequality. It is also well-known that a “strict” inequality between operators does not imply strict monotonicity of the spectral radii in general—some additional structure is required. We present a number of sufficient conditions on both the cone and the operators for such a strict ordering to hold which generalise known results in the literature, and have utility in comparison arguments, ubiquitous in positive systems theory.     

Real Interpolation of Compact Bilinear Operators

We establish an analog for bilinear operators of the compactness interpolation result for bounded linear operators proved by Cwikel and Cobos, Kühn and Schonbek. We work with the assumption that \(T:(A_0+A_1) \times (B_0+B_1) \longrightarrow E_0+E_1\) is bounded, but we also study the case when this does not hold. Applications are given to compactness of convolution operators and compactness of commutators of bilinear Calderón–Zygmund operators.     

Sturm-Liouville operators in the half axis with local perturbations

We give conditions which imply equivalence of the Lebesgue measure with respect to a measure μ generated as an average of spectral measures corresponding to Sturm-Liouville operators in the half axis. We apply this to prove that some spectral properties of these operators hold for large sets of boundary conditions if and only if they hold for large sets of positive local perturbations.     

Stability of discrete-time positive evolution operators on ordered Banach spaces and applications

In this paper we discuss stability problems for a class of discrete-time evolution operators generated by linear positive operators acting on certain ordered Banach spaces. Our approach is based upon a new representation result that links a positive operator with the adjoint operator of its restriction to a Hilbert subspace formed by sequences of Hilbert–Schmidt operators. This class includes the evolution operators involved in stability and optimal control problems for linear discrete-time stochastic systems. The inclusion is strict because, following the results of Choi, we have proved that there are positive operators on spaces of linear, bounded and self-adjoint operators which have not the representation that characterize the completely positive operators. As applications, we introduce a new concept of weak-detectability for pairs of positive operators, which we use to derive sufficient conditions for the existence of global and stabilizing solutions for a class of generalized discrete-time Riccati equations. Finally, assuming weak-detectability conditions and using the method of Lyapunov equations we derive a new stability criterion for positive evolution operators.     

On the iterates of positive linear operators

We have devised a new method for the study of the asymptotic behavior of the iterates of positive linear operators. This technique enlarges the class of operators for which the limit of the iterates can be computed.     

A Note on Property(ω)

In this note we study the property(ω),a variant of Weyl's theorem introduced by Rakoevic,by means of the new spectrum.We establish for a bounded linear operator defined on a Banach space a necessary and sufficient condition for which both property(ω) and approximate Weyl's theorem hold.As a consequence of the main result,we study the property(ω) and approximate Weyl's theorem for a class of operators which we call the λ-weak-H(p) operators.     

On the best constant in Hyers–Ulam stability of some positive linear operators

We obtain the infimum of the Hyers–Ulam stability constants for Stancu, Bernstein and Kantorovich operators and prove that in a class of certain positive linear operators this infimum for Bernstein operator has a minimality property.     

An estimate for some positive linear operators in L p spaces

In this note we prove that an estimate due to Ivanov and Pichugov, for positive linear convolution operators in L _p spaces, can be extended to a class of positive linear operators which are not of convolution type.     

On statistical approximation of a general class of positive linear operators extended in q-calculus

In this paper we present a general class of positive linear operators of discrete type based on q-calculus and we investigate their weighted statistical approximation properties by using a Bohman–Korovkin type theorem. We also mark out two particular cases of this general class of operators.     

Ordinary differential operators in Hilbert spaces and Fredholm pairs

Let be a path of bounded operators on a real Hilbert space, hyperbolic at . We study the Fredholm theory of the operator . We relate the Fredholm property of to the stable and unstable linear spaces of the associated system . Several examples are included to point out the differences with respect to the finite dimensional case, in particular concerning the role of the spectral flow. We define a general class of paths A for which many properties typical of the finite dimensional framework still hold. Our motivation is to develop the linear theory which is necessary for the set-up of Morse homology on Hilbert manifolds. Received: 9 March 2001; in final form: 1 March 2002 / Published online: 2 December 2002     

性质$(\omega)$的注解

In this note we study the property （ω）, a variant of Weyl＇s theorem introduced by Rakocevic, by means of the new spectrum. We establish for a bounded linear operator defined on a Banach space a necessary and sufficient condition for which both property （ω） and approximate Weyl＇s theorem hold. As a consequence of the main result, we study the property （ω） and approximate Weyl＇s theorem for a class of operators which we call the λ-weak-H（p） operators.     

(ω)性质及Weyl型定理

(ω)性质是Rakocevic给出的Weyl定理的一种变化.本文通过定义新的谱集,给出了有界线性算子同时满足(ω)性质和a-Weyl定理的充要条件.同时,利用所得的主要结论,研究了H(p)算子的(ω)性质.     

Minimization of a functional over the set of causal operators of causal Hilbert space

The minimization problem for a quadratic functional defined on the set of nonwarning (causal) operators acting in a causal Hilbert space can be regarded as an abstrat analog of the Wiener problem on constructing the optimal nonwarning filter. A similar problem also arises in the linear control problem with the quadratic performance criterion (in this case the transfer operators of a closed control system serve as causal ones). The introduction of causal operators in filtering theory and control theory is a mathematical expression of the causality principle, which must be taken into account for a number of problems. In the present paper we attempt to systematize the mathematical foundations of the abstract linear filtering theory, for which its basic results are expressed in terms of operators describing the filtering problem. We introduce and study a class of finite operators, a natural generalization of the class of causal operators, and give a solution of the minimization problem for a quadratic positive functional defined on the set of causal operators acting in a “discrete” causal space. Bibliography: 54 titles. Translated fromProblemy Matematicheskogo Analiza, No. 14, 1995. pp. 143–187.     

A pointwise estimate for positive dyadic shifts and some applications

We prove a (sharp) pointwise estimate for positive dyadic shifts of complexity m which is linear in the complexity. This can be used to give a pointwise estimate for Calderón-Zygmund operators and to answer a question originally posed by Lerner. Several applications to weighted estimates for both multilinear Calderón-Zygmund operators and square functions are discussed.     

New Korovkin Type Theorem for Non-Tensor Meyer–König and Zeller Operators

In this paper, we introduce a certain class of linear positive operators via a generating function, which includes the non-tensor MKZ operators and their non-trivial extension. In investigating the approximation properties, we prove a new Korovkin type approximation theorem by using appropriate test functions. We compute the rate of convergence of these operators by means of the modulus of continuity and the elements of modified Lipschitz class functions. Furthermore, we give functional partial differential equations for this class. Using the corresponding equations, we calculate the first few moments of the non-tensor MKZ operators and investigate their approximation properties. Finally, we state the multivariate versions of the results and obtain the convergence properties of the multivariate Meyer–König and Zeller operators.     

On Positive Hilbert–Schmidt Operators

We study classes of self-adjoint Hilbert–Schmidt operators, focusing on sufficient conditions for the operators to be positive. The integral kernels for which the conditions hold true encompass kernel functions that arise in the setting of elliptic Calogero–Moser type integrable N-particle systems, a context where the positivity property has crucial consequences.     

Local Lp-saturation of positive linear convolution operators

The approximation of unbounded functions by positive linear operators under multiplier enlargement is investigated. It is shown that a very wide class of positive linear operators can be used to approximate functions with arbitrary growth on the real line. Estimates are given in terms of the usual quantities which appear in the Shisha-Mond theorem. Examples are provided.     

Iso-Huygens Deformations of Homogeneous Differential Operators Related to a Special Cone of Rank $${\text{3}}$$

We consider iso-Huygens deformations of homogeneous hyperbolic Gindikin operators related to a special cone of rank ${\text{3}}$ . The deformations are carried out with the use of Stellmacher--Lagnese and Calogero--Moser potentials. Using the notion of gauge equivalence of operators and the algebraic method of intertwining operators, we write out the fundamental solutions of the deformed operators in closed form and give sufficient conditions for the Huygens principle to hold for these operators in the strengthened or ordinary form.     

Properties and applications of $$P_n$$-simple functionals

In this paper some representations of a linear positive functional are established. These results are extended for linear positive operators \(L_n:C[a,b] \rightarrow C[a,b]\). Also, approximation properties of linear positive operators, expressed in terms of moduli of smoothness, are considered. In the last section, the remainder term in various approximation processes is studied.