Search results for "Operator"
showing 10 items of 1427 documents
Dissipative operators and differential equations on Banach spaces
1991
If we consider the initial value problem Inline Equation $$x'(t) = f(t,x(t)),{\text{ }}x(0) = {x_0}$$ on the real line, it is well known that one—sided bounds like Inline Equation $$\left[ {f(t,x) - f\left( {t,y} \right)} \right]\left( {x - {\text{y}}} \right) \leqslant \omega {\left( {x - y} \right)^2}$$ give much better information about the behaviour of solutions than the Lipschitz- type estimates Inline Equation $$ \left| {f\left( {t,x} \right) - f\left( {t,y} \right)} \right| \leqslant L\left| {x - y} \right|,$$ because ω, unlike L, may be negative.
Spectral Asymptotics for $$\mathcal {P}\mathcal {T}$$ Symmetric Operators
2019
\(\mathcal {P}\mathcal {T}\)-symmetry has been proposed as an alternative to self-adjointness in quantum physics, see Bender et al. (J Math Phys 40(5):2201–2229, 1999), Bender and Mannheim (Phys Lett A 374(15–16):1616–1620, 2010). Thus for instance, if we consider a Schrodinger operator on Rn, $$\displaystyle P=-h^2\Delta +V(x), $$ the usual assumption of self-adjointness (implying that the potential V is real valued) can be replaced by that of \(\mathcal {P}\mathcal {T}\)-symmetry: $$\displaystyle V\circ \iota =\overline {V}, $$ where ι : Rn →Rn is an isometry with ι2 = 1≠ι. If we introduce the parity operator \(\mathcal {P}_\iota u(x)=u(\iota (x))\) and the time reversal operator \(\mathc…
Approximation Operators of Binomial Type
1999
Our objective is to present a unified theory of the approximation operators of binomial type by exploiting the main technique of the so- called “ umbral calculus” or “finite operator calculus” (see [18], [20]-[22]). Let us consider the basic sequence (bn)n≥0 associated to a certain delta operator Q. By supposing that b n (x) ≥ 0, x ∈ [0, ∞), our purpose is to put in evidence some approximation properties of the linear positive operators (L Q n ) n≥1 which are defined on C[0,1] by \( L_n^Qf = \sum\limits_{k = 0}^n {\beta _n^Q{,_k}f\left( {\frac{k}{n}} \right),\beta _{n{,_k}}^Q\left( x \right): = } \frac{1}{{{b_n}\left( n \right)}}\left( {\begin{array}{*{20}{c}} n \\ k \end{array}} \right){b_…
Explicit solutions of two-point boundary value operator problems
1988
Soit H un espace de Hilbert, complexe, separable et soit L(H) l'algebre de tous les operateurs lineaires bornes sur H. On etudie des conditions d'existence non triviales pour le probleme aux valeurs limites operateurs: t 2 X (2) +tA 1 X (1) +A 0 X=0; M 11 X(a)+N 11 X(b)+M 12 X (1) (a)+N 12 X (1) (b)=0, M 21 X(a)+N 21 X(b)+M 22 X (1) (a)+N 22 X (1) (b)=0, 0<a≤t≤b ou M ij , N ij , pour 1≤i, j≤2 et A 0 , A 1 sont des operateurs de L(H). Sous certaines hypotheses concernant l'existence des solutions d'une equation operateur algebrique X 2 +B 1 X+B 0 =0, on obtient des solutions explicites au probleme aux limites
The Variation of the Fractional Maximal Function of a Radial Function
2017
Abstract In this article, we study the regularity of the non-centered fractional maximal operator $M_{\beta}$. As the main result, we prove that there exists $C(n,\beta)$ such that if $q=n/(n-\beta)$ and $f$ is radial function, then $\|DM_{\beta}f\|_{L^{q}({\mathbb{R}^n})}\leq C(n,\beta)\|Df\|_{L^{1}({\mathbb{R}^n})}$. The corresponding result was previously known only if $n=1$ or $\beta=0$. Our proofs are almost free from one-dimensional arguments. Therefore, we believe that the new approach may be very useful when trying to extend the result for all $f\in W^{1,1}({\mathbb{R}^n})$.
The best constant for the Sobolev trace embedding from into
2004
Abstract In this paper we study the best constant, λ 1 ( Ω ) for the trace map from W 1 , 1 ( Ω ) into L 1 ( ∂ Ω ) . We show that this constant is attained in BV ( Ω ) when λ 1 ( Ω ) 1 . Moreover, we prove that this constant can be obtained as limit when p ↘ 1 of the best constant of W 1 , p ( Ω ) ↪ L p ( ∂ Ω ) . To perform the proofs we will look at Neumann problems involving the 1-Laplacian, Δ 1 ( u ) = div ( Du / | Du | ) .
On the Unit Ball of Operator-valued H 2-functions
2009
Let X be a complex Banach space and let H 2 (\( \mathbb{D} \), X) denote the space of X-valued analytic functions in the unit disc such that $$ \mathop {sup}\limits_{0 < r < 1} \int_0^{2\pi } {\left\| {F\left( {re^{it} } \right)} \right\|^2 \frac{{dt}} {{2\pi }} < \infty .} $$ It is shown that a function F belongs to the unit ball of H 2 ( \( \mathbb{D} \), X) if and only if there exist f∈H ∞ (\( \mathbb{D} \), X) and ϕ∈H ∞ (\( \mathbb{D} \)) such that $$ \left\| {f\left( z \right)} \right\|^2 + \left| {\varphi \left( z \right)} \right|^2 \leqslant 1 and F\left( z \right) = \frac{{f\left( z \right)}} {{1 - z\varphi \left( z \right)}} $$ for |z| < 1.
Weak commutation relations of unbounded operators and applications
2011
Four possible definitions of the commutation relation $[S,T]=\Id$ of two closable unbounded operators $S,T$ are compared. The {\em weak} sense of this commutator is given in terms of the inner product of the Hilbert space $\H$ where the operators act. Some consequences on the existence of eigenvectors of two number-like operators are derived and the partial O*-algebra generated by $S,T$ is studied. Some applications are also considered.
Variational differential inclusions without ellipticity condition
2020
The paper sets forth a new type of variational problem without any ellipticity or monotonicity condition. A prototype is a differential inclusion whose driving operator is the competing weighted $(p,q)$-Laplacian $-\Delta_p u+\mu\Delta_q u$ with $\mu\in \mathbb{R}$. Local and nonlocal boundary value problems fitting into this nonstandard setting are examined.
A model of internal and external competition in a High Speed Rail line
2015
This paper is a contribution to evaluate structural and behavioral changes in railway passenger markets. The novel elements of our analysis are the following: (i) the consideration of inter-modal and intra-modal competition, (ii) the presence of public and private operators, and (iii) endogenous service frequency. After calibrating the model using actual data from two Spanish High Speed Rail lines, simulation exercises allow us to conclude the following. Privatization, whether entry occurs or not, would prompt an increase in prices and a reduction in the number of train services, eventually leading to welfare decreases, as compared with a regime where the incumbent rail operator remained pu…