Existence of Positive Solutions for Operator Equations and Applications to Semipositone Problems

In this paper we study the existence of positive solutions of the following operator equation in a Banach space E: where G(x, λ) = λKFx+e₀, K: E↦ E is a linear completely continuous operator, F: P↦ E is a nonlinear continuous , bounded operator, e₀∈ E, λ is a parameter and P is a cone of Banach space E. Since F is not assumed to be positive and e₀ may be a negative element, the operator equation is a so-called semipositone problem. We prove that under certain super-linear conditions on the operator F the operator equation has at least one positive solution for λ > 0 sufficiently small, and that under certain sub-linear conditions on the operator F the operator equation has at least one positive solution for λ > 0 sufficiently large. In addition, we briefly outline an application of our results which simplify previous theorems in the literature.     

Study of the Free Streaming Operator in Slab Geometry in Dependence of the Boundary Conditions

We study the free streaming operator T_∧ in a slab domain with boundary conditions described by a linear matrix operator Λ acting between the ‘incoming’ and ‘outgoing’ particle fluxes. Under suitable assumptions on the entries of Λ, it is proved that the resolvent operator of T_∧ is positive. It is proved also that T_∧ is the generator of a positive strongly continuous semigroup, whose type depends on the norm of the entries of Λ. Some examples are given. In particular the case of Maxwell type boundary conditions is studied and the location of the spectrum of T_∧ is improved. © 1997 by B. G. Teubner Stuttgart – John Wiley & Sons Ltd. Math. Meth. Appl. Sci., Vol. 20, 717–736 (1997).     

A Sturm–Liouville problem depending rationally on the eigenvalue parameter

We consider the Sturm–Liouville problem (1.1) and (1.2) with a potential depending rationally on the eigenvalue parameter. With these equations a λ ‐linear eigenvalue problem is associated in such a way that L₂‐solutions of (1.1), (1.2) correspond to eigenvectors of a linear operator. If the functions q and u are real and satisfy some additional conditions, the corresponding linear operator is a definitizable self‐adjoint operator in some Krein space. Moreover we consider the problem (1.1) and (1.3) on the positive half‐axis. Here we use results on the absense of positive eigenvalues for Sturm–Liouville operators to exclude critical points of the associated definitizable operator. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Some results on b-AM-compact operators

We show that for positive operator B : E → E on Banach lattices, if there exists a positive operator S : E → E such that:1.SB ≤ BS;2.S is quasinilpotent at some x₀ > 0; 3.S dominates a non-zero b-AM-compact operator, then B has a non-trivial closed invariant subspace. Also, we prove that for two commuting non-zero positive operators on Banach lattices, if one of them is quasinilpotent at a non-zero positive vector and the other dominates a non-zero b-AM-compact operator, then both of them have a common non-trivial closed invariant ideal. Then we introduce the class of b-AM-compact-friendly operators and show that a non-zero positive b-AM- compact-friendly operator which is quasinilpotent at some x₀ > 0 has a non-trivial closed invariant ideal.     

Range Inclusions and Approximate Source Conditions with General Benchmark Functions

The paper is devoted to the analysis of ill-posed operator equations Ax = y with injective linear operator A and solution x ₀ in a Hilbert space setting. We present some new ideas and results for finding convergence rates in Tikhonov regularization based on the concept of approximate source conditions by means of using distance functions with a general benchmark. For the case of compact operator A and benchmark functions of power-type, we can show that there is a one-to-one correspondence between the maximal power-type decay rate of the distance function and the best possible Hölder exponent for the noise-free convergence rate in Tikhonov regularization. As is well-known, this exponent coincides with the supremum of exponents in power-type source conditions. The main theorem of this paper is devoted to the impact of range inclusions under the smoothness assumption that x ₀ is in the range of some positive self-adjoint operator G. It generalizes a convergence rate result proven for compact G in Hofmann and Yamamoto (Inverse Problems 2005; 21:805–820) to the case of general operators G with nonclosed range.     

On subinvariant elements in Banach lattices

In the paper we obtain an ergodic decomposition generated by a positive operator T:E → E (E being a Banach lattice having the projection property). If T is a power bounded operator and if E is a KB-space, then the decomposition permits us to describe sufficient conditions for the existence of nonzero T-subinvariant elements in E. We also study the relationship between the Hopf decomposition in Banach lattices and the one defined in the paper in order to obtain sufficient conditions for the existence of a T-invariant weak order unit.     

Some Applications of the Regularity and Irreducibility on Transport Theory

This paper is concerned with the spectral analysis of transport operator with general boundary conditions in L ₁-setting. This problem will be investigated under results from the theory of positive linear operators, irreducibility and regularity of the collision operator. The basic problems treated here are notions of essential spectra, spectral bound and leading eigenvalues.     

On the Peripheral Spectrum of Order Continuous,Positive Operators

Let E be an order complete Banach function lattice and T a positive, eventually compact, order continuous operator on E. We study necessary conditions under which the peripheral spectrum of T is fully cyclic in terms of certain bands of the underlying Banach function lattice E. A set of sufficient conditions is also given. Examples are presented to demonstrate our methods.     

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.     

Sampling of Band-Limited Vectors

Given a self-adjoint, positive definite operator on a Hilbert space the concept of band-limited vectors (with a given band-width) is developed, using the spectral decomposition of that operator. By means of this concept sufficient conditions on collections of linear functionals {j_n}\{\varphi_{\nu}\} are derived which imply that all band limited vectors in a given class are uniquely determined resp.can be reconstructed in a stable way from the set of discrete values {j_n(f)}\{\varphi_{\nu}(f)\}.     

Zeros of Fredholm Operator Valued Hp–Functions

This paper is concerned with Fredholm operator valued H^p – functions on the unit disc, where the Fredholm operators action a Banach space. Sufficient conditions are presented which guarantee that Fatou's theorem is valid. Using the theory of traces and determinants on quasi – Banach operator ideals, we develop conditions that guarantee that the zeros of Fredholm operator valued H^p – functions satisfy the Blaschke condition.     

Positive Absolutely Summing Operators on the Köthe Spaces

In this paper we characterize the positive absolutely summing operators on the Köthe space E(X), with X a Banach lattice, extending a previous result. We prove that a composition operator of two positive absolutely summing operators is a positive absolutely summing operator. An interpolation result for the positive absolutely summing operators is obtained.     

Asymptotic behavior of a differential operator with discontinuities at two points

In this paper, we study a Sturm–Liouville operator with eigenparameter‐dependent boundary conditions and transmission conditions at two interior points. By establishing a new operator A associated with the problem, we prove that the operator A is self‐adjoint in an appropriate space H, discuss completeness of its eigenfunctions in H, and obtain its Green function. Copyright © 2010 John Wiley & Sons, Ltd.     

Around Operator Monotone Functions

We show that the symmetrized product AB + BA of two positive operators A and B is positive if and only if f(A+B) ￡ f(A)+f(B){f(A+B)\leq f(A)+f(B)} for all non-negative operator monotone functions f on [0,∞) and deduce an operator inequality. We also give a necessary and sufficient condition for that the composition f °g{f \circ g} of an operator convex function f on [0,∞) and a non-negative operator monotone function g on an interval (a, b) is operator monotone and present some applications.     

A relationship between two almost f-algebra products

Let A be an uniformly complete almost f-algebra. Then is a positively generated ordered vector subspace of A with as a positive cone. If is a positive linear operator, we put the linear operator defined by with for all is the algebra of all order bounded linear operators of A). Let denote the range of and let's define a new product by putting for all . It is easily checked that if then , this shows that if it happens that the product is associative then A is an almost f- algebra with respect to this new product. It turns out that a necessarily and sufficient condition in order that be an associative product is that is a commutative subalgebra of . We find necessarily and sufficient conditions on T in order that is an almost f-algebra (respect.; d-algebra, f-algebra) product. Such conditions are described in terms of the algebraic and order structure of the algebra .?The converse problem is also studied. More precisely, let A be an uniformly complete almost f-algebra and assume that is another almost f-algebra product on A. The aim is to find sufficient conditions in order that there exist such that for all . It will be showed that a sufficient condition is that A is a d-algebra with respect to the initial product. An example is produced which shows that the condition "A is a d-algebra with respect to the initial product" can not be weakened. Received November 8, 1999; accepted in final form February 14, 2000.     

Bushell's equations and polar decompositions

We show that for any real number t with t ≠ ±1, every invertible operator M on a Hilbert space admits a new polar decomposition M = PUP^–t where P is positive definite and U is unitary, and that the corresponding polar map is homeomorphism. The positive definite factor P of M appears as the negative square root of the unique positive definite solution of the nonlinear operator equation X^t = M * XM. This extends the classical matrix and operator polar decomposition when t = 0. For t = ± 1, it is shown that the positive definite solution sets of X^±1 = M * XM form geodesic submanifolds of the Banach–Finsler manifold of positive definite operators and coincide with fixed point sets of certain non‐expansive mappings, respectively (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Parafermion vertex operator algebras

This paper reviews some recent results on the parafermion vertex operator algebra associated to the integrable highest weight module L(k, 0) of positive integer level k for any affine Kac-Moody Lie algebra ĝ, where g is a finite dimensional simple Lie algebra. In particular, the generators and the C ₂-cofiniteness of the parafermion vertex operator algebras are discussed. A proof of the well-known fact that the parafermion vertex operator algebra can be realized as the commutant of a lattice vertex operator algebra in L(k, 0) is also given.     

Dobrushin Coefficients of Ergodicity and Asymptotically Stable L 1-Contractions

We extend an inequality (which involves the Dobrushin coefficient of ergodicity; see Cohen et al.⁽⁴⁾) to any linear bounded operator with domain and codomain L ¹-spaces. We use the extended Dobrushin coefficient of ergodicity, that appears in the inequality, in order to obtain sufficient conditions for the uniform asymptotic stability of a positive contraction of an L ¹-space. We conclude the paper by studying a class of strongly asymptotically stable positive contractions.     

Applications of Wigner’s theorem to positive maps preserving norm of operator products

We apply Wigner’s theorem to positive maps on standard operator algebras that preserve norm of operator products or sum of singular values of operator products. It follows that such preservers are of the form ϕ(A) = U AU* with U either a unitary or antiunitary operator. Selected from Journal of Mathematical Study, 2004, 37(1): 11–16     

A Fredholm Determinant Formula for Section Determinants of Bounded Operators

We show that the section determinant of e^A can be expressed, under certain conditions, by the Fredholm determinant of an integral operator. The kernel function of this integral operator is computed explicitly in terms of the operator A. As a simple consequence we derive a Weierstrass type product expansion for the section determinant.