Characterization of unbounded spectral operators with spectrum in a half-line

LetT be a possibly unbounded linear operator in the Banach spaceX such thatR(t)=(t+T)^?1 is defined onR ⁺. LetS=TR(I?TR) and letB(.,.) denote the Beta function. Theorem 1.1.T is a scalar-type spectral operator with spectrum in [0, ∞) if and only if $$sup\left\{ {B\left( {k,k} \right)^{ - 1} \int_0^\infty {\left| {x*S^k \left( t \right)x} \right|{{dt} \mathord{\left/ {\vphantom {{dt} t}} \right. \kern-\nulldelimiterspace} t};\left\| x \right\| \leqslant 1,} \left\| {x*} \right\| \leqslant 1,k \geqslant 1} \right\}< \infty .$$ A “local” version of this result is formulated in Theorem 2.2.     

A Quantitative Version of the Bishop–Phelps Theorem for Operators in Hilbert Spaces

In this paper, with the help of spectral integral, we show a quantitative version of the Bishop-Phelps theorem for operators in complex Hilbert spaces. Precisely, let H be a complex Hilbert space and 0 ε 1/2. Then for every bounded linear operator T : H → H and x0 ∈ H with ||T|| = 1 = ||x0|| such that ||Tx0|| 1 ε, there exist xε∈ H and a bounded linear operator S : H → H with||S|| = 1 = ||xε|| such that ||Sxε|| = 1, ||xε-x0|| ≤ (2ε)1/2 + 4(2ε)1/2, ||S-T|| ≤(2ε)1/2.     

On the similarity problem for polynomially bounded operators on hilbert space

Partial solutions are obtained to Halmos’ problem, whether or not any polynomially bounded operator on a Hilbert spaceH is similar to a contraction. Central use is made of Paulsen’s necessary and sufficient condition, which permits one to obtain bounds on ‖S‖ ‖S ^?1‖, whereS is the similarity. A natural example of a polynomially bounded operator appears in the theory of Hankel matrices, defining $$R_f = \left( {\begin{array}{*{20}c} {S*} \\ 0 \\ \end{array} \begin{array}{*{20}c} {\Gamma _f } \\ S \\ \end{array} } \right)$$ onl ² ⊕l ², whereS is the shift and Γ _f the Hankel operator determined byf withf′ ∈ BMOA. Using Paulsen’s condition, we prove thatR _f is similar to a contraction. In the general case, combining Grothendieck’s theorem and techniques from complex function theory, we are able to get in the finite dimensional case the estimate $$\left\| S \right\|\left\| {S^{ - 1} } \right\| \leqq M^4 log(dim H)$$ whereSTS ^?1 is a contraction and assuming \(\left\| {p\left( T \right)} \right\| \leqq M\left\| p \right\|_\infty \) wheneverp is an analytic polynomial on the disc.     

Strong duality for infinite-dimensional vector-valued programming problems

LetX,Y andZ be locally convex real topological vector spaces,A?X a convex subset, and letC?Y,E?Z be cones. Letf:X→Z beE-concave andg:X→Y beC-concave functions. We consider a concave programming problem with respect to an abstract cone and its strong dual problem as follows: $$\begin{gathered} (P)maximize f(x), subject to x \in A, g(x) \in C, \hfill \\ (SD)minimize \left\{ {\mathop \cup \limits_{\varphi \in C^ + } \max \{ (f + \varphi \circ g)(A):E\} } \right\}, \hfill \\ \end{gathered} $$ , whereC ⁺ denotes the set of all nonnegative continuous linear operators fromY toZ and (SD) is the strong dual problem to (P). In this paper, the authors find a necessary condition of strong saddle point for Problem (P) and establish the strong duality relationships between Problems (P) and (SD).     

A grothendieck factorization theorem on 2-convex schatten spaces

We prove that for every bounded linear operatorT:C ^2p →H(1≤p<∞,H is a Hilbert space,C ^{2 p} p is the Schatten space) there exists a continuous linear formf onC ^p such thatf≥0, ‖f‖(C _C ^p)*=1 and $$\forall x \in C^{2p} , \left\| {T(x)} \right\| \leqslant 2\sqrt 2 \left\| T \right\|< f\frac{{x * x + xx*}}{2} > 1/2$$ . Forp=∞ this non-commutative analogue of Grothendieck’s theorem was first proved by G. Pisier. In the above statement the Schatten spaceC ^2p can be replaced byE _E ² whereE ⁽²⁾ is the 2-convexification of the symmetric sequence spaceE, andf is a continuous linear form onC _E. The statement can also be extended toL _E{(su2)}(M, τ) whereM is a Von Neumann algebra,τ a trace onM, E a symmetric function space.     

A minimax theorem for vector-valued functions,part 2

We deal with the minimax problem relative to a vector-valued functionf: X ₀×Y ₀ »V, where a partial ordering in the topological vector spaceV is induced by a closed and convex coneC. In Ref. 1, under suitable hypotheses, we proved that $$Max\bigcup\limits_{s\varepsilon X_0 } {Min_w f(s,Y_0 )} \subset Min\bigcup\limits_{t\varepsilon Y_0 } {Maxf(X_0 ,t) + C;}$$ the exact meaning of the symbols is given in Section 2. In this work, we prove that, under a reasonable setting of hypotheses, the previous inclusion holds and also we have that $$Min_w \bigcup\limits_{t\varepsilon Y_0 } {Max} f(X_0 ,t) \subset Max\bigcup\limits_{s\varepsilon X_0 } {Min_w } f(s,Y_0 ) - C.$$      

Error bounds for the finite element approximation of a degenerate quasilinear parabolic variational inequality

In this paper, we establish some error bounds for the continuous piecewise linear finite element approximation of the following problem: Let Ω be an open set in ? ^d , withd=1 or 2. GivenT>0,p ∈ (1, ∞),f andu ₀; findu ∈K, whereK is a closed convex subset of the Sobolev spaceW ₀ ^1,p (Ω), such that for anyv∈K $$\begin{gathered} \int\limits_\Omega {u_1 (\upsilon - u) dx + } \int\limits_\Omega {\left| {\nabla u} \right|^{p - 2} } \nabla u \cdot \nabla (\upsilon - u) dx \geqslant \int\limits_\Omega {f(\upsilon - u) dx for} a.e. t \in (0,T], \hfill \\ u = 0 on \partial \Omega \times (0,T] and u(0,x) = u_0 (x) for x \in \Omega . \hfill \\ \end{gathered} $$ We prove error bounds in energy type norms for the fully discrete approximation using the backward Euler time discretisation. In some notable cases, these error bounds converge at the optimal rate with respect to the space discretisation, provided the solutionu is sufficiently regular.     

A strong law of large numbers for nonexpansive vector-valued stochastic processes

A mapT: X→X on a normed linear space is callednonexpansive if ‖Tx-Ty‖≤‖x-y‖∀x, y∈X. Let (Ω, Σ,P) be a probability space, an increasing chain of σ-fields spanning Σ,X a Banach space, andT: X→X. A sequence (x_n) of strongly -measurable and stronglyP-integrable functions on Ω taking on values inX is called aT-martingale if . LetT: H→H be a nonexpansive mapping on a Hilbert spaceH and let (x_n) be aT-martingale taking on values inH. If then x _n /n converges a.e. LetT: X→X be a nonexpansive mapping on ap-uniformly smooth Banach spaceX, 1<p≤2, and let (x_n) be aT-martingale (taking on values inX). If then there exists a continuous linear functionalf∈X ^* of norm 1 such that If, in addition, the spaceX is strictly convex, x _n /n converges weakly; and if the norm ofX ^* is Fréchet differentiable (away from zero), x _n /n converges strongly. This work was supported by National Science Foundation Grant MCS-82-02093     

On the numerical integration of nonlinear initial value problems by linear multistep methods

We study the numerical solution of the nonlinear initial value problem $$\left\{ {\begin{array}{*{20}c} {{{du(t)} \mathord{\left/ {\vphantom {{du(t)} {dt}}} \right. \kern-\nulldelimiterspace} {dt}} + Au(t) = f(t),t > 0} \\ {u(0) = c,} \\ \end{array} } \right.$$ whereA is a nonlinear operator in a real Hilbert space. The problem is discretized using linear multistep methods, and we assume that their stability regions have nonempty interiors. We give sharp bounds for the global error by relating the stability region of the method to the monotonicity properties ofA. In particular we study the case whereAu is the gradient of a convex functional φ(u).     

An inequality associated with the uncertainty principle

We prove \(\left\| F \right\|_{2,\Omega } \leqslant c({\rm T} \Omega )\left\| f \right\|_{A{}_T} \) , whereF is the Fourier transform off,||F||_2,Ω is theL ²-norm ofF on \([ - \Omega ,\Omega ],\left\| f \right\|_{A{}_T} \) is the absolutely convergent Fourier series norm for 2T-periodic functions, and $$c(T\Omega ) = (\frac{1}{\pi }\int\limits_{ - T\Omega }^{T\Omega } {\frac{{\sin ^2 \gamma }}{{\gamma ^2 }}d\gamma } )^{1/2} $$ Analogous inequalities, depending on prolate spheroidal wave functions, are more difficult to prove and their constants are less explicit.     

Sharp weak-type inequalities for Hilbert transform and Riesz projection

The paper is devoted to the study of the weak norms of the classical operators in the vector-valued setting.

1. Let S, H denote the singular integral involution operator and the Hilbert transform on $L^p \left( {\mathbb{T}, \ell _\mathbb{C}^2 } \right)$ , respectively. Then for 1 ≤ p ≤ 2 and any f, $$\left\| {\mathcal{S}f} \right\|_{p,\infty } \leqslant \left( {\frac{1} {\pi }\int_{ - \infty }^\infty {\frac{{\left| {\tfrac{2} {\pi }\log \left| t \right|} \right|^p }} {{t^2 + 1}}dt} } \right)^{ - 1/p} \left\| f \right\|p,$$ $$\left\| {\mathcal{H}f} \right\|_{p,\infty } \leqslant \left( {\frac{1} {\pi }\int_{ - \infty }^\infty {\frac{{\left| {\tfrac{2} {\pi }\log \left| t \right|} \right|^p }} {{t^2 + 1}}dt} } \right)^{ - 1/p} \left\| f \right\|p.$$ Both inequalities are sharp.
2. Let P ₊ and P _? stand for the Riesz projection and the co-analytic projection on $L^p \left( {\mathbb{T}, \ell _\mathbb{C}^2 } \right)$ , respectively. Then for 1 ≤ p ≤ 2 and any f, $$\left\| {P + f} \right\|_{p,\infty } \leqslant \left\| f \right\|_p ,$$ $$\left\| {P - f} \right\|_{p,\infty } \leqslant \left\| f \right\|_p .$$ Both inequalities are sharp.
3. We establish the sharp versions of the estimates above in the nonperiodic case.

The results are new even if the operators act on complex-valued functions. The proof rests on the construction of an appropriate plurisubharmonic function and probabilistic techniques.     

Minimization of convex functionals involving nested maxima: Nonconcave duality and algorithms

We consider problems of the form $$\mathop {\min }\limits_u J(u) + \sum\limits_{0 \leqslant i \leqslant n} {a_i } \mathop {\max }\limits_{0 \leqslant j \leqslant i} \Theta _j (u),$$ in which thea′ _i s are nonnegative numbers, andJ and the Θ _j 's are convex functionals on a reflexive Banach spaceU. We show that such problems may arise, in particular, when scheduling investments over several periods of time. By expressing the nested maximizations in a recursive way, we transform the above problem into that of finding the minimax of some functional Φ(u, α), where α ranges over the unit cube of ? ⁿ . Although the dependence of Φ on α is neither linear nor even concave, we show that a saddle point does exist for this problem. Moreover, we propose a very simple dual algorithm to solve max_α min _u Φ(u, α), whose convergence is proved and whose limit yields the true optimum, although the dual functional is not concave and does have local maxima in general. Another algorithm with this same property, but without convergence proof, is also proposed. Finally, a numerical example illustrates the use of these approaches and algorithms.     

Some properties of elliptic Riesz means at critical index on Hp(TH)

Suppose f∈H^p(Tⁿ), 0 _r ^δ , δ=n/p?(n+1)/2. In this paper we eastablish the following inequality $$\mathop {\sup }\limits_{R > 1} \left\{ {\frac{1}{{\log R}}\int_1^R {\left\| {\sigma _r^\delta } \right\|_{H^p (T^R )}^p \frac{{dr}}{r}} } \right\}^{1/p} \leqslant C_{R,p} \left\| f \right\|_{H^p (T^R )} $$ It implies that $$\mathop {\lim }\limits_{R \to \infty } \frac{1}{{\log R}}\int_1^R {\left\| {\sigma _r^\delta - f} \right\|_{H^p (T^R )}^p \frac{{dr}}{r}} = 0$$ Moreover we obtain the same conclusion when p=1 and n=1.     

Explicit solutions of infinite quadratic programs

LetH be a Hilbert space,X be a real Banach space,A: H→X be an operator withD (A) dense inH, G: H→H be positive definite,x ∈D (A) andb ∈H. Consider the quadratic programming problem: $$\begin{gathered} QP:Minimize \frac{1}{2}\left\langle {p,x} \right\rangle + \left\langle {x,Gx} \right\rangle \hfill \\ subject to Ax = b \hfill \\ \end{gathered} $$ In this paper, we obtain an explicit solution to teh above problem using generalized inverses.     

Spectral Shorted Operators

If $$\mathcal{H}$$ is a Hilbert space, $$\mathcal{S}$$ is a closed subspace of $$\mathcal{H},$$ and A is a positive bounded linear operator on $$\mathcal{H},$$ the spectral shorted operator $$\rho \left( {\mathcal{S},\mathcal{A}} \right)$$ is defined as the infimum of the sequence $$\sum (\mathcal{S},A^n )^{1/n} ,$$ where denotes $$\sum \left( {\mathcal{S},B} \right)$$ the shorted operator of B to $$\mathcal{S}.$$ We characterize the left spectral resolution of $$\rho \left( {\mathcal{S},\mathcal{A}} \right)$$ and show several properties of this operator, particularly in the case that dim $${\mathcal{S} = 1.}$$ We use these results to generalize the concept of Kolmogorov complexity for the infinite dimensional case and for non invertible operators.     

Vector optimization problems with quasiconvex constraints

Let X be a real linear space, a convex set, Y and Z topological real linear spaces. The constrained optimization problem min _C f(x), is considered, where f : X ₀→ Y and g : X ₀→ Z are given (nonsmooth) functions, and and are closed convex cones. The weakly efficient solutions (w-minimizers) of this problem are investigated. When g obeys quasiconvex properties, first-order necessary and first-order sufficient optimality conditions in terms of Dini directional derivatives are obtained. In the special case of problems with pseudoconvex data it is shown that these conditions characterize the global w-minimizers and generalize known results from convex vector programming. The obtained results are applied to the special case of problems with finite dimensional image spaces and ordering cones the positive orthants, in particular to scalar problems with quasiconvex constraints. It is shown, that the quasiconvexity of the constraints allows to formulate the optimality conditions using the more simple single valued Dini derivatives instead of the set valued ones.      

Second order linear Volterra integrodifferential equations

In this paper we study mild and classical solutions of the second order linear Volterra integrodifferential equation $$(VE^f )\left\{ {\begin{array}{*{20}c} {u''(t) = Au(t) + {\text{ }}\int_0^t {B(t - s)u(s)ds + f(t){\text{ }}for{\text{ }}t \in [0,T]} } \\ {u(0) = x{\text{ }}and{\text{ }}u'(0) = y,} \\ \end{array} } \right.$$ whereA is a closed linear operator whose domainD(A) is not necessarily dense in a Banach spaceX, and {B(t)|t≥0} is a family of bounded linear operators from the Banach space,D(A) endowed with the graph norm intoX. We also give two examples to illustrate the abstract results.     

A Sharp Remez Inequality for Trigonometric Polynomials

We obtain a sharp Remez inequality for the trigonometric polynomial T _n of degree n on [0,2π): $$\|T_n \|_{L_\infty([0,2\pi))} \le \biggl(1+2\tan^2 \frac{n\beta}{4m} \biggr) { \|T_n \|_{L_\infty ([0,2\pi) \setminus B )}}, $$ where $\frac{2\pi}{m}$ is the minimal period of T _n and $|B|=\beta<\frac {2\pi m}{n}$ is a measurable subset of $\mathbb {T}$ . In particular, this gives the asymptotics of the sharp constant in the Remez inequality: for a fixed n, $$\mathcal{C}(n, \beta)=1+ \frac{(n\beta)^2}{8} +O \bigl(\beta^4\bigr), \quad\beta \to0, $$ where $$\mathcal{C}(n,\beta):= \sup_{|B|=\beta}\sup_{T_n} \frac{ \|T_n \|_{L_\infty([0,2\pi ))}}{ \|T_n \|_{L_\infty ([0,2\pi) \setminus B )}}. $$ We also obtain sharp Nikol’skii’s inequalities for the Lorentz spaces and net spaces. Multidimensional variants of Remez and Nikol’skii’s inequalities are investigated.     

Two Parameter Asymptotic Spectra in the Uniformly Elliptic Case

In this article we study the abstract two parameter eigenvalue problem $$\begin{gathered} T_1 u_1 = \left( {\lambda _1 V_{11} + \lambda _2 V_{12} } \right)u_1 , \left\| {u_1 } \right\| = 1 \hfill \\ T_2 u_2 = \left( {\lambda _1 V_{21} + \lambda _2 V_{22} } \right)u_2 , \left\| {u_2 } \right\| = 1 \hfill \\ \end{gathered}$$ where, in the Hilbert spaces H_j, T_j is self-adjoint, bounded below and has compact resolvent, and V_jk are self-adjoint bounded operators, (?1)^j+kV_jk >> 0, j, k = 1, 2. An eigenvalue λ for this problem is a point in R² satisfying both equations. Under appropriate conditions, the eigenvalues λⁿ = (λ₁ ⁿ, λ₂ ⁿ) are countable and in R². We aim to describe the set of limit points of λⁿ/∥λⁿ∥, as ∥λⁿ∥ → ∞, in terms of the V_jk.