共查询到20条相似文献,搜索用时 375 毫秒
1.
A (stochastic) operator-theoretic approach leads to expresssions for inverses of linear and nonlinear stochastic operators—useful for the solution of linear or nonlinear stochastic differential equations. Operator equations are developed for inverses of linear or nonlinear stochastic operators. Series expressions are obtained which allow writing the solution y=?1x of the operator equation y=x. Special cases are studied in which may be linear or nonlinear, deterministic or stochastic in various combinations. 相似文献
2.
Generalized almost periodic solutions and ergodic properties of quasi-autonomous dissipative systems
Alain Haraux 《Journal of Differential Equations》1983,48(2):269-279
Let H be a real Hilbert space, A a maximal monotone operator in H and a measurable function which is S1-almost periodic. Assuming that the positive trajectories of the equation are bounded (in H) for t ? 0, we construct a kind of generalized almost periodic “solution” of the equation, and we show how to deduce information of ergodic type on the asymptotic behavior of the trajectories as t → +∞. 相似文献
3.
B. D. Gel’man 《Functional Analysis and Its Applications》2016,50(1):31-38
A new approach to nonstandard boundary value problems is suggested. For such problems, we construct equivalent inclusions with surjective operators and study the solvability of these inclusions. The paper consists of two parts. The first part deals with problems in which the right-hand side of the equation is a Lipschitz mapping (Section 3); in the second part (Section 4), this mapping is completely continuous with respect to a surjective operator A. The paper also gives examples of how our theorems can be applied when studying nonstandard boundary value problems. 相似文献
4.
A. Jiménez-Vargas J.M. Sepulcre Moisés Villegas-Vallecillos 《Journal of Mathematical Analysis and Applications》2014
We introduce the notion of Lipschitz compact (weakly compact, finite-rank, approximable) operators from a pointed metric space X into a Banach space E. We prove that every strongly Lipschitz p-nuclear operator is Lipschitz compact and every strongly Lipschitz p-integral operator is Lipschitz weakly compact. A theory of Lipschitz compact (weakly compact, finite-rank) operators which closely parallels the theory for linear operators is developed. In terms of the Lipschitz transpose map of a Lipschitz operator, we state Lipschitz versions of Schauder type theorems on the (weak) compactness of the adjoint of a (weakly) compact linear operator. 相似文献
5.
C.A Stuart 《Journal of Functional Analysis》1980,38(2):169-187
For a bounded analytic function, ?, on the unit disk, denote the operators of multiplication by ? on H2(?D) and L2(?D), respectively. In their 1973 paper, Deddens and Wong asked whether there is an analytic Toeplitz operator that commutes with a nonzero compact operator, and whether every operator that commutes with an analytic Toeplitz operator has an extension that commutes with the corresponding multiplication operator on L2. In the first part of this paper, we give an explicit example of an analytic Toeplitz operator Tφ that settles both of these questions. This operator commutes with a nonzero compact operator (a composition operator followed by an analytic Toeplitz operator). The only operators in the commutant of Tφ that extend to commute with Mφ are analytic Toeplitz operators. Although the commutant of Tφ contains more than just analytic Toeplitz operators, Tφ is irreducible. The remainder of the paper seeks to explain more fully the phenomena incorporated in this example by introducing a class of analytic functions, including the function φ, and giving additional conditions on functions g in the class to determine whether Tg commutes with nonzero compact operators, whether Tg is irreducible, and which operators in the commutant of Tg extend to the commutant of Mg. In particular, we find representations for operators in the commutant and second commutant of Tg. 相似文献
6.
New first-order conformally covariant differential operators Pk on spinor-k-forms, i.e., tensor products of contravariant spinors with k-forms, in an arbitrary n-dimensional pseudo-Riemannian spin manifold, are introduced. This provides a series of generalizations of the Dirac operator , in analogy with the series of generalizations (introduced by the author in [1]) of the Maxwell operator and the conformally covariant Laplacian on functions. In particular, new intertwining operators for representations of SU(2, 2) and SO(p + 1, q + 1) are found. Related nonlinear covariant operators are also introduced, and mixed nonlinear covariant systems are obtained by coupling to the Yang-Mills-Higgs-Dirac system in dimension 4. The spinor-form bundle is isomorphic with E(3) = E ? E ? E, where E is the spin bundle, and the Pk give a covariant operator on sections of E(3). This is generalized to a covariant operator on E(2l + 1). The relation of powers of these operators to higher-order covariant operators on lower spin bundles (analogous to the relation between and ) is discussed. 相似文献
7.
Barry Simon 《Journal of Functional Analysis》1981,42(3):347-355
We consider Schrödinger operators for a large class of potentials. V. We show that if H? = E? has a polynomially bounded solution ? then E is in the spectrum of H. This is accomplished by proving that the spectrum of H as an operator on L2 is identical to its spectrum as an operator on the weighted L2 space, L2δ. 相似文献
8.
Hans G Kaper Gary K Leaf Arthur J Lindeman 《Journal of Mathematical Analysis and Applications》1975,50(1):42-65
The one-group neutron transport equation is commonly given as an integrodifferential equation for the neutron density ψ(x, ω) over a domain G × S in the five-dimensional phase space . In this paper we show how, by decomposing the domain of the transport operator into a complementary pair of manifolds by means of a projection operator, any transport problem can be formulated, on either manifold, in terms of a symmetric positive definite operator. We use Friedrichs' method to extend the operator to a selfadjoint operator and look for a generalized solution by minimizing a certain functional over the appropriate Hilbert space. A Ritz-Galerkin type approximation procedure is formulated, and an estimate for the difference between the exact and approximate solution is given. The procedure is illustrated for a special choice of finite dimensional subspace. 相似文献
9.
Let Ω be a bounded Lipschitz domain in ? n , n ? 2, and let L be a second-order matrix strongly elliptic operator in Ω written in divergence form. There is a vast literature dealing with the study of domains of fractional powers of operators corresponding to various problems (beginning with the Dirichlet and Neumann problems) with homogeneous boundary conditions for the equation Lu = f, including the solution of the Kato square root problem, which arose in 1961. Mixed problems and a class of problems for higher-order systems have been covered as well. We suggest a new abstract approach to the topic, which permits one to obtain the results that we deem to be most important in a much simpler and unified way and cover new operators, namely, classical boundary operators on the Lipschitz boundary Γ = ?Ω or part of it. To this end, we simultaneously consider two well-known operators associated with the boundary value problem. 相似文献
10.
Ronald E Bruck 《Journal of Mathematical Analysis and Applications》1980,76(1):159-173
We show that if u is a bounded solution on + of u″(t) ?Au(t) + f(t), where A is a maximal monotone operator on a real Hilbert space H and f∈Lloc2(+;H) is periodic, then there exists a periodic solution ω of the differential equation such that u(t) ? ω(t) 0 and u′(t) ? ω′(t) → 0 as t → ∞. We also show that the two-point boundary value problem for this equation has a unique solution for boundary values in and that a smoothing effect takes place. 相似文献
11.
Steven G. Krantz 《Journal of Functional Analysis》1979,34(3):456-471
It is known that a function on n which can be well approximated by polynomials, in the mean over Euclidean balls, is Lipschitz smooth in the usual sense. In this paper an analogous theorem is proved in which n is replaced by a set X, the averages over balls are replaced by a family of sublinear operators satisfying certain axioms, and the polynomials are replaced by a class of functions having certain regularity properties with respect to the averaging operators. Applications are given to function theory on domains in n, to nilpotent Lie groups, and to the classical Euclidean case. The first application provides a characterization of the duals of Hardy spaces on the ball in n. 相似文献
12.
Richard A Alò André de Korvin Vo Van Tho 《Journal of Mathematical Analysis and Applications》1977,61(1):72-96
We develop in this article a strong nonlinear integral and obtain a Riesz-type theorem (utilizing this integral) for the class of (nonlinear) Hammerstein operators. The integral is extended to the class E() of E-valued totally -measurable functions and convergence theorems are studied. Then an exchange of information is carried out between the operators and the corresponding set functions; for example, the implication of the operator being compact or unconditionally summing is drawn. In the latter case it is shown that the representing set function is analogous to strongly bounded set functions. A vast body of literature exists for both of these concepts. 相似文献
13.
Palle E.T Jørgensen 《Journal of Functional Analysis》1975,20(2):105-135
In this paper we apply the theory of second-order partial differential operators with nonnegative characteristic form to representations of Lie groups. We are concerned with a continuous representation U of a Lie group G in a Banach space . Let be the enveloping algebra of G, and let dU be the infinitesimal homomorphism of into operators with the Gårding vectors as a common invariant domain. We study elements in of the form with the Xj,'s in the Lie algebra .If the elements X0, X1,…, Xr generate as a Lie algebra then we show that the space of C∞-vectors for U is precisely equal to the C∞-vectors for the closure . This result is applied to obtain estimates for differential operators.The operator is the infinitesimal generator of a strongly continuous semigroup of operators in . If X0 = 0 we show that this semigroup can be analytically continued to complex time ζ with Re ζ > 0. The generalized heat kernels of these semigroups are computed. A space of rapidly decreasing functions on G is introduced in order to treat the heat kernels.For unitary representations we show essential self-adjointness of all operators with X0 in the real linear span of the Xj's. An application to quantum field theory is given.Finally, the new characterization of the C∞-vectors is applied to a construction of a counterexample to a conjecture on exponentiation of operator Lie algebras.Our results on semigroups of exponential growth, and on the space of C∞ vectors for a group representation can be viewed as generalizations of various results due to Nelson-Stinespring [18], and Poulsen [19], who prove essential self-adjointness and a priori estimates, respectively, for the sum of the squares of elements in a basis for (the Laplace operator). The work of Hörmander [11] and Bony [3] on degenerate-elliptic (hypoelliptic) operators supplies the technical basis for this generalization. The important feature is that elliptic regularity is too crude a tool for controlling commutators. With the aid of the above-mentioned hypoellipticity results we are able to “control” the (finite dimensional) Lie algebra generated by a given set of differential operators. 相似文献
14.
Tomas P Schonbek 《Journal of Mathematical Analysis and Applications》1977,58(3):527-540
Let be a Banach space; S and T bounded scalar-type operators in . Define Δ on the space of bounded operators on by ΔX = TX ? XS if X is a bounded operator. We set up a calculus for Δ which allows us to consider f(Δ), for f a complex-valued bounded Borel measurable function on the spectrum of Δ, as an operator in the space of bounded operators whose domain is a subspace of operators which we call measure generating. This calculus is used to obtain some results on when the kernel of Δ is a complemented subspace of the space of bounded operators on . 相似文献
15.
Allan C. Peterson 《Journal of Mathematical Analysis and Applications》1975,52(3):573-582
In this paper we are concerned with proving comparison theorems, under various assumptions, for the (p, q)-boundary value problem for the nth order nonlinear differential equation and the linear differential equation lny = (?1)qk(t)y, where ln is the classical nth order linear operator with leading coefficient one. 相似文献
16.
Gilles Pisier 《Journal of Functional Analysis》1978,29(3):397-415
We study a conjecture of Grothendieck on bilinear forms on a C1-algebra . We prove that every “approximable” operator from into 1 factors through a Hilbert space, and we describe the factorization. In the commutative case, this is known as Grothendieck's theorem. These results enable us to prove a conjecture of Ringrose on operators on a C1-algebra. In the Appendix, we present a new proof of Grothendieck's inequality which gives an improved upper bound for the so-called Grothendieck constant kG. 相似文献
17.
Lawrence Corwin 《Journal of Functional Analysis》1981,44(3):328-347
Let N be a nilpotent Lie group and Q a tempered distribution on N. We say that Q is a left -multiplier if convolution on the left by Q takes Schwartz class functions to Schwartz class functions; there is a similar definition for right -multipliers. We show that if ? is an irreducible unitary representation of N, then one can define ρ(Q):∞:(ρ)→∞:(ρ) whenever Q is a left -multiplier. The main results of the paper characterize left -multipliers Q on Heisenberg groups in terms of the transform operators ?(Q) and show how this characterization can be used to find fundamental solutions of some left invariant differential operators. There is also an example of a left -multiplier which is not a right -multiplier. 相似文献
18.
E.B Davies 《Journal of Differential Equations》1985,60(1):103-130
We obtain a positive lower bound to the spectrum of certain second-order elliptic operators H on N has a Lipschitz boundary and the coefficients of H become singular as one approaches the boundary. We also find a general formula for the order of the asymptotic eigenvalue distribution of H in some situations where the classical limit formula of Weyl, Courant, Titchmarsh, and others is not applicable. 相似文献
19.
Reuven Meidan 《Journal of Mathematical Analysis and Applications》1977,61(2):552-563
In this work nonlinear translation-varying operators are analyzed and represented by means of a generalized impulse response. This is the response of the transpose operator to the family of shifted impulse functionals. Continuous operators from a topological vector space into the space of functions on Rn, as well as -bounded operators, are investigated. 相似文献
20.
R Lowen 《Journal of Mathematical Analysis and Applications》1978,64(2):446-454
Starting from a defining differential equation of the kernel of an exponential operator with normalization ∫?∞∞W(λ, t, u) du = 1, we determine Sλ for various p(t) including; for example, p(t) a quadratic polynomial, all the known exponential operators are recovered and some new ones are constructed. It is shown that all the exponential operators are approximation operators. Further approximation properties of these operators are discussed. For example, functions satisfying are characterized. Several results of C. P. May are also improved. 相似文献