COMPOSITION OPERATORS FROM p-BLOCH SPACES TO LITTLE q-BLOCH SPACES ON THE UNIT BALL OF Cn

For all 0 p, q ∞, let C denote the composition operator from q-Bloch spaces βp to little p-Bloch spaces β0q on the unit ball of Cn. In this article, necessary and sufficient conditions for C to be a bounded or compact operator are given.     

Gauge-periodic point perturbations on the Lobachevsky plane

We study periodic point perturbations of the Shrödinger operator with a uniform magnetic field on the Lobachevsky plane. We prove that the spectrum gaps of the perturbed operator are labeled by the elements of the K₀ group of aC ^* algebra associated with the operator. In particular, if theC ^* algebra has the Kadison property, then the operator spectrum has a band structure.     

Exponential asymptotic property of a parallel repairable system with warm standby under common-cause failure

We discuss exponential asymptotic property of the solution of a parallel repairable system with warm standby under common-cause failure. This system can be described by a group of partial differential equations with integral boundary. First we show that the positive contraction C₀-semigroup T(t) [Weiwei Hu, Asymptotic stability analysis of a parallel repairable system with warm standby under common-cause failure, Acta Anal. Funct. Appl. 8 (1) (2006) 5-20] which is generated by the operator corresponding to these equations is a quasi-compact operator. Then by using [Weiwei Hu, Asymptotic stability analysis of a parallel repairable system with warm standby under common-cause failure, Acta Anal. Funct. Appl. 8 (1) (2006) 5-20] that 0 is an eigenvalue of the operator with algebraic index one and the C₀-semigroup T(t) is contraction, we conclude that the spectral bound of the operator is zero. By using the above results the exponential asymptotical stability of the time-dependent solution of the system follows easily.     

A further characteristic of abstract convexity structures on topological spaces

In this paper, we give a characteristic of abstract convexity structures on topological spaces with selection property. We show that if a convexity structure C defined on a topological space has the weak selection property then C satisfies H₀-condition. Moreover, in a compact convex subset of a topological space with convexity structure, the weak selection property implies the fixed point property.     

ON THE SPECTRAL RADIUS OF IRREDUCIBLE AND WEAKLY IRREDUCIBLE OPERATORS IN BANACH LATPICES

Abstract

If T is an operator on a Banach lattice E we call T weakly irreducible if E contains no non-trivial T-invariant bands. We prove that if E is order complete and if the weakly irreducible operator T > 0 is in (E′_oo ? E)^⊥⊥ then T has positive spectral radéus. Prom this follows that Jentesch's theorem holds in arbitrary Banach function spaces.

If [Ttilde] denotes the restriction of T′ to E′_oo, 0 ? T an order continuous operator, then T is weakly irreducible if and only if [Ttilde]: E′_oo→E′_oo is weakly irreducible.

Finally we show that the majorizing, irreducible operator T ≥ 0, has positive spectral radius if either Tⁿ is weakly compact or E has property (P) or T is strongly majorizing.     

A short proof of the Arendt-Chernoff-Kato theorem

We give a short new proof of the Arendt-Chernoff-Kato theorem, which characterizes generators of positive C ₀ semigroups in order unit spaces. The proof avoids half-norms and subdifferentials and is based on a sufficient condition for an operator to have positive inverse, which is new even for matrices.     

Resolvent characterisation of generators of cosine functions and C 0-groups

The paper provides new characterisations of generators of cosine functions and C ₀-groups on UMD spaces and their applications to some classical problems in cosine function theory. In particular, we show that on UMD spaces, generators of cosine functions and C ₀-groups can be characterised by means of a complex inversion formula. This allows us to provide a strikingly elementary proof of Fattorini’s result on square root reduction for cosine function generators on UMD spaces. Moreover, we give a cosine function analogue of McIntosh’s characterisation of the boundedness of the H ^∞ functional calculus for sectorial operators in terms of square function estimates. Another result says that the class of cosine function generators on a Hilbert space is exactly the class of operators which possess a dilation to a multiplication operator on a vector-valued L ₂ space. Finally, we prove a cosine function analogue of the Gomilko-Feng-Shi characterisation of C ₀-semigroup generators and apply it to answer in the affirmative a question by Fattorini on the growth bounds of perturbed cosine functions on Hilbert spaces.     

Commutators, C0-semigroups and resolvent estimates

We study the existence and the continuity properties of the boundary values on the real axis of the resolvent of a self-adjoint operator H in the framework of the conjugate operator method initiated by Mourre. We allow the conjugate operator A to be the generator of a C₀-semigroup (finer estimates require A to be maximal symmetric) and we consider situations where the first commutator [H,iA] is not comparable to H. The applications include the spectral theory of zero mass quantum field models.     

Random matrices and subexponential operator spaces

We introduce and study a generalization of the notion of exact operator space that we call subexponential. Using Random Matrices we show that the factorization results of Grothendieck type that are known in the exact case all extend to the subexponential case, but we exhibit (a continuum of distinct) examples of non-exact subexponential operator spaces, as well as a C*-algebra that is subexponential with constant 1 but not exact. We also show that OH, R + C and max(?₂) (or any other maximal operator space) are not subexponential.     

Abstract potential operator and spectral methods for a class of degenerate evolution problems

The spectral theory of operators in Banach spaces is employed to treat a class of degenerate evolution equations. A basic role is played by the assumption that the Banach space under consideration may be expressed as a direct sum of two suitable subspaces. Two methods for solving the problem are studied. The first method is based on the expansion of the resolvent of a closed operator into Laurent series in a neighbourhood of 0. The second one makes use of the theory of abstract potential operators. In particular, an extension of the Hille-Yosida theorem on infinitesimal generators of (C₀) semigroups of linear operators is obtained. Some examples relative to operators appearing in many applications to partial differential equations are given.     

Isomorphism properties of Toeplitz operators and pseudo-differential operators between modulation spaces

We investigate the lifting property of modulation spaces and construct explicit isomorphisms between them. For each weight function ?? and suitable window function ??, the Toeplitz operator (or localization operator) Tp _?? (??) is an isomorphism from $M_{({\omega _0})}^{p,q}$ , onto $M_{({\omega _0}/\omega )}^{p,q}$ for every p, q ?? [1,??] and arbitrary weight function ?? ₀. The methods involve the pseudo-differential calculus of Bony and Chemin and the Wiener algebra property of certain symbol classes of pseudo-differential Operators.     

Some properties of root functions of ordinary second-order differential operators in the absence of the carleman condition

On a finite interval G of the real line, we consider the root functions of an ordinary second-order differential operator without any boundary conditions for the case in which the imaginary part of the spectral parameter is unbounded.We refine the estimates for the C-and L _p -norms of a root function and its first derivative on a compact set contained in the interior of G for the case in which the Carleman condition fails.A sufficient condition is obtained for the root functions of an ordinary second-order differential operator to have the Bessel property, assuming that the Carleman condition fails. We show that, under certain conditions, this problem can be reduced to analyzing the Bessel property of systems of exponentials.     

Forward-Backward and Tseng’s Type Penalty Schemes for Monotone Inclusion Problems

We deal with monotone inclusion problems of the form 0 ∈ A x + D x + N _C (x) in real Hilbert spaces, where A is a maximally monotone operator, D a cocoercive operator and C the nonempty set of zeros of another cocoercive operator. We propose a forward-backward penalty algorithm for solving this problem which extends the one proposed by Attouch et al. (SIAM J. Optim. 21(4): 1251-1274, 2011). The condition which guarantees the weak ergodic convergence of the sequence of iterates generated by the proposed scheme is formulated by means of the Fitzpatrick function associated to the maximally monotone operator that describes the set C. In the second part we introduce a forward-backward-forward algorithm for monotone inclusion problems having the same structure, but this time by replacing the cocoercivity hypotheses with Lipschitz continuity conditions. The latter penalty type algorithm opens the gate to handle monotone inclusion problems with more complicated structures, for instance, involving compositions of maximally monotone operators with linear continuous ones.     

A semigroup approach to the Gnedenko system with single vacation of a repairman

We investigate the Gnedenko system with one repairman who can take vacations. Our main focus is on the time asymptotic behaviour of the system. Using C ₀-semigroup theory for linear operators we first prove the well-posedness of the system and the existence of a unique positive dynamic solution given an initial value. Then by analysing the spectral distribution of the system operator and taking into account the irreducibility of the semigroup generated by the system operator we show that the dynamic solution converges strongly to the steady state solution. Thus we obtain asymptotic stability of the dynamic solution.     

The mixed boundary value problem, Krein resolvent formulas and spectral asymptotic estimates

For a second-order symmetric strongly elliptic operator A on a smooth bounded open set in Rn, the mixed problem is defined by a Neumann-type condition on a part Σ₊ of the boundary and a Dirichlet condition on the other part Σ₋. We show a Kre?n resolvent formula, where the difference between its resolvent and the Dirichlet resolvent is expressed in terms of operators acting on Sobolev spaces over Σ₊. This is used to obtain a new Weyl-type spectral asymptotics formula for the resolvent difference (where upper estimates were known before), namely , where C0,+ is proportional to the area of Σ₊, in the case where A is principally equal to the Laplacian.     

New Lorentz spaces for the restricted weak-type Hardy's inequalities

Associated to the class of restricted weak-type weights for the Hardy operator R_p, we find a new class of Lorentz spaces for which the normability property holds. This result is analogous to the characterization given by Sawyer for the classical Lorentz spaces. We also show that these new spaces are very natural to study the existence of equivalent norms described in terms of the maximal function.     

Integral mappings between Banach spaces

We consider the classes of “Grothendieck-integral” (G-integral) and “Pietsch-integral” (P-integral) linear and multilinear operators (see definitions below), and we prove that a multilinear operator between Banach spaces is G-integral (resp. P-integral) if and only if its linearization is G-integral (resp. P-integral) on the injective tensor product of the spaces, together with some related results concerning certain canonically associated linear operators. As an application we give a new proof of a result on the Radon-Nikodym property of the dual of the injective tensor product of Banach spaces. Moreover, we give a simple proof of a characterization of the G-integral operators on C(K,X) spaces and we also give a partial characterization of P-integral operators on C(K,X) spaces.     

The generalized Roper-Suffridge extension operator in Banach spaces (II)

In this paper, we generalize the Roper-Suffridge extension operator from Cn to Banach spaces. It is proved that this operator preserves the biholomorphic ? starlikeness on some domains in Banach spaces. From these, we may construct a lots of concrete examples about biholomorphic ? starlike mappings on some domains Ω in Cn, or Hilbert spaces, or Banach spaces from univalent ? starlike functions on the unit disc U in C. Meanwhile, the growth theorems of the corresponding mappings are given. Some results of Gong and Liu, Roper and Suffridge, Graham et al. in Cn are extended to Hilbert spaces or Banach spaces.     

Property (w) and perturbations III

The property (w) is a variant of Weyl's theorem, for a bounded operator T acting on a Banach space. In this note we consider the preservation of property (w) under a finite rank perturbation commuting with T, whenever T is polaroid, or T has analytical core K(λ₀I−T)={0} for some λ₀∈C. The preservation of property (w) is also studied under commuting nilpotent or under injective quasi-nilpotent perturbations. The theory is exemplified in the case of some special classes of operators.