Generalized Resolvents of Symmetric Operators

Mathematical Notes -     

The Uniqueness Problem for Subordinate Resolvents with Potential Theoretical Methods

We present an analytic version of the following uniqueness problem for Markov processes: if two subprocesses of a given transient, Borel right process have the same excessive functions then they coincide. Our treatment is given in terms of subordinate sub-Markovian resolvents of kernels in the sense of P. A. Mayer and uses essentially the subordination operators introduced by G. Mokobodzki.      

Effective Rates of Convergence for the Resolvents of Accretive Operators

We extract explicit, computable, and highly uniform rates for the strong convergence of the resolvents of set-valued, m-accretive, and uniformly accretive at zero/?-expansive operators on general real Banach spaces to the zero of each operator. This is achieved through proof mining on the proof of a theorem by García–Falset the motivation of which originates from a classical work by Reich. For the bound extraction we make use of a modulus of accretivity at zero, a notion introduced recently by Kohlenbach and the author, as well as a modulus of ?-expansivity, a notion introduced analogously here.     

Bounded Local Resolvents for Operators with Single-Valued Extension Property

Let T be a bounded operator with (SVEP) on its localizable spectrum \(\sigma _\mathrm{loc}(T)\). We show that for every open subset U of \(\sigma _\mathrm{loc}(T)\), there exists a unit vector x whose local spectrum coincides with the closure of U, and such that its local resolvent function is bounded. This result answers positively to an open question stated by several authors, and extends the both cases of operators with trivial divisible subspace and operators whose point spectrum has empty interior.     

Smooth Measures and Regular Strongly Supermedian Kernels Generating Sub-Markovian Resolvents

In the context of a transient Borel right Markov process with a fixed excessive measure , we characterize the regular strongly supermedian kernels, producing smooth measures by the Revuz correspondence. In the case of the measures charging no -semipolar sets, this is the analytical counterpart of a probabilistic result of Revuz, Fukushima, and Getoor and Fitzsimmons, concerning the positive continuous additive functionals. We also consider the case of the measures charging no set that is both -polar and -negligible (U being the potential part of ), answering to a problem of Revuz.     

Banach空间中闭线性算子广义预解式存在定理

在Banach空间中研究闭线性算子广义逆扰动问题和广义预解式存在性问题.给出了闭线性算子广义逆在T-有界扰动下的一些稳定特征,这些特征推广了在有界线性算子情形、闭线性算子有界扰动情形以及闭线性算子保值域或保核空间情形的一些已知结果.以此为基础,得到了闭线性算子广义预解式存在的一些充要条件及其广义预解式的显式表达式.作为应用,给出了闭Fredholm算子和闭半-Fredholm算子的广义预解式存在性特征.     

锥线性算子的延拓定理

建立了一种锥分离定理,据此证明了锥线性算子的延拓定理,作为应用,给出了正线性算子的延拓定理。     

On Bounded Local Resolvents

It is known that each normal operator on a Hilbert space with nonempty interior of the spectrum admits vectors with bounded local resolvent. We generalize this result for Banach space operators with the decomposition property (δ) (in particular for decomposable operators). Moreover, the same result holds for operators with interior points in the localizable spectrum.     

Resolvents of elliptic cone operators

We prove the existence of sectors of minimal growth for general closed extensions of elliptic cone operators under natural ellipticity conditions. This is achieved by the construction of a suitable parametrix and reduction to the boundary. Special attention is devoted to the clarification of the analytic structure of the resolvent.     

On Smooth Local Resolvents

We exhibit an example of a bounded linear operator on a Banach space which admits an everywhere defined local resolvent with continuous derivatives of all orders.     

Nonlinear Kernels and Their Resolvents

In [1], Dellacherie showed that also nonlinear kernels can have a resolvent associated with them. In this text, we will generalise some of his results to the case where the complete maximum principle does not hold.     

The Method of Alternating Resolvents Revisited

The purpose of this article is to prove a strong convergence result associated with a generalization of the method of alternating resolvents introduced by the authors in convergence of the method of alternating resolvents [4 O. A. Boikanyo and G. Moro?anu ( to appear ). Strong convergence of the method of alternating resolvents . J. Nonlinear Convex Anal.  [Google Scholar]] under minimal assumptions on the control parameters involved. Thus, this article represents a significant improvement of the article mentioned above.     

Resolvents of operators and partial functional differential equations with nonautonomous past

To a backward evolution family on a Banach space X we associate an abstract differential operator G through the integral equation on a Banach space of X-valued functions on . We compute the resolvent of the restriction of this operator to a smaller domain to obtain a generator. We then apply the results to prove existence, exponential stability and exponential dichotomy of solutions to partial functional equations with nonautonomous past as discussed in [S. Brendle, R. Nagel, Dist. Contin. Dynam. Systems 8 (2002) 953-966]. Our main tools are spectral mapping theorems for evolution semigroups and hyperbolicity criteria.     

EXTENSION OF SMOOTHING FUNCTIONS TO SYMMETRIC CONE COMPLEMENTARITY PROBLEMS

The paper uses Euclidean Jordan algebras as a basic tool to extend smoothing functions, which include the Chen-Mangasarian class and the Fischer-Burmeister smoothing functions, to symmetric cone complementarity problems. Computable formulas for these functions and their Jacobians are derived. In addition, it is shown that these functions are Lipschitz continuous with respect to parameter # and continuously differentiable on J × J for any μ 〉 0.     

Resolvents in Semi-linear Harmonic Spaces

We will show that the semi-linear potential kernels defined in [4] have a resolvent associated with them. Furthermore, the bounded excessive functions of this resolvent correspond to the bounded hyperharmonic functions as they do in linear potential theory.     

Extension of primal-dual interior point methods to diff-convex problems on symmetric cones

We consider the extension of primal dual interior point methods for linear programming on symmetric cones, to a wider class of problems that includes approximate necessary optimality conditions for functions expressible as the difference of two convex functions of a special form. Our analysis applies the Jordan-algebraic approach to symmetric cones. As the basic method is local, we apply the idea of the filter method for a globalization strategy.