共查询到20条相似文献,搜索用时 15 毫秒
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Bernhard H. Haak Peer Christian Kunstmann 《Integral Equations and Operator Theory》2006,55(4):497-533
We study linear systems, described by operators A, B, C for which the state space X is a Banach space.We suppose that − A generates a bounded analytic semigroup and give conditions for admissibility of B and C corresponding to those in G. Weiss’ conjecture. The crucial assumptions on A are boundedness of an H∞-calculus or suitable square function estimates, allowing to use techniques recently developed by N. Kalton and L. Weis. For observation spaces Y or control spaces U that are not Hilbert spaces we are led to a notion of admissibility extending previous considerations by C. Le Merdy. We also obtain a characterisation of wellposedness for the full system. We give several examples for admissible operators
including point observation and point control. At the end we study a heat equation in X = Lp(Ω), 1 < p < ∞, with boundary observation and control and prove its wellposedness for several function spaces Y and U on the boundary ∂Ω. 相似文献
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A continuous quadratic form (“quadratic form”, in short) on a Banach space X is: (a) delta-semidefinite (i.e., representable as a difference of two nonnegative quadratic forms) if and only if the corresponding symmetric linear
operator factors through a Hilbert space; (b) delta-convex (i.e., representable as a difference of two continuous convex functions) if and only if T is a UMD-operator. It follows, for instance, that each quadratic form on an infinite-dimensional L
p
(μ) space (1 ≤ p ≤ ∞) is: (a) delta-semidefinite iff p ≥ 2; (b) delta-convex iff p > 1. Some other related results concerning delta-convexity are proved and some open probms are stated.
The first author was supported by NSF grant DMS-0555670. The second author was supported by the Russian Foundation for Basic
Research, Grant 05-01-00066, and by Grant NSh-5813.2006.1. The third author was supported in part by the Ministero dell’Università
e della Ricerca of Italy. 相似文献
4.
The Bohl-Bohr-Amerio-Kadets theorem states that the indefinite integral y=Pφ of an almost periodic (ap) is again ap if y is bounded and the Banach space X does not contain a subspace isomorphic to c0. This is here generalized in several directions: Instead of it holds also for φ defined only on a half-line , instead of ap functions abstract classes with suitable properties are admissible, can be weakened to φ in some “mean” class , then ; here contains all f∈L1loc with in for all h>0 (usually strictly); furthermore, instead of boundedness of y mean boundedness, y in some , or in , ergodic functions, suffices. The Loomis-Doss result on the almost periodicity of a bounded Ψ for which all differences Ψ(t+h)−Ψ(t) are ap for h>0 is extended analogously, also to higher order differences. Studying “difference spaces” in this connection, we obtain decompositions of the form: Any bounded measurable function is the sum of a bounded ergodic function and the indefinite integral of a bounded ergodic function. The Bohr-Neugebauer result on the almost periodicity of bounded solutions y of linear differential equations P(D)y=φ of degree m with ap φ is extended similarly for ; then provided, for example, y is in some with U=L∞ or is totally ergodic and, for the half-line, Reλ?0 for all eigenvalues P(λ)=0. Analogous results hold for systems of linear differential equations. Special case: φ bounded and Pφ ergodic implies Pφ bounded. If all Reλ>0, there exists a unique solution y growing not too fast; this y is in if , for quite general . 相似文献
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Maria Alessandra Ragusa 《NoDEA : Nonlinear Differential Equations and Applications》2007,13(5-6):605-617
In the paper the author proves properties of existence, uniqueness and regularity for divergence form elliptic equations,
extending these results from the linear case to the quasilinear one. 相似文献
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We study four transformations which lead from one well-posed linear
system to another: time-inversion, flow^-inversion, time-flow-inversion and
duality. Time-inversion means reversing the direction of time, flow-inversion
means interchanging inputs with outputs, while time-flow-inversion means doing
both of the inversions mentioned before. A well-posed linear system is
time-invertible if and only if its operator semigroup extends to a group. The
system is flow-invertible if and only if its input-output map has a bounded
inverse on some (hence, on every) finite time interval [0, ] ( > 0). This is
true if and only if the transfer function of has a uniformly bounded inverse
on some right half-plane. The system is time-flow-invertible if and only if
on some (hence, on every) finite time interval [0, ], the combined operator
from the initial state and the input function to the final state and the output
function is invertible. This is the case, for example, if the system is conservative,
since then is unitary. Time-flow-inversion can sometimes, but not
always, be reduced to a combination of time- and flow-inversion. We derive a
surprising necessary and sufficient condition for to be time-flow-invertible:
its system operator must have a uniformly bounded inverse on some left halfplane.
Finally, the duality transformation is always possible.We show by some
examples that none of these transformations preserves regularity in general.
However, the duality transformation does preserve weak regularity. For all
the transformed systems mentioned above, we give formulas for their system
operators, transfer functions and, in the regular case and under additional
assumptions, for their generating operators. 相似文献
7.
In this paper, we investigate whether the 124 nonsingular toric Fano
4-folds admit totally nondegenerate embeddings from abelian surfaces or
not. In consequence, we determine the possibilities of these embeddings,
except for the remaining 18 nonsingular toric Fano 4-folds.
Received: 12 July 2002 相似文献
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Differential inclusions of a retarded type with a small real parameter >0 in part of the derivatives are considered. We prove upper semicontinuity of the map set of solutions at =0+ inC[0, 1]×(L
2(0, 1)–weak) topology. In case of constant delay lower semicontinuity inC[0, 1]×(L
1(0, 1)–strong) is shown. 相似文献
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We study when a Banach space with absolute norm may have polynomial numerical indices equal to one. In the real case, we show that a Banach space X with absolute norm, which has the Radon-Nikodým property or is Asplund, satisfies n(2)(X)<1 unless it is one-dimensional. In the complex case, we show that the only Banach spaces X with absolute norm and the Radon-Nikodým property which satisfy n(2)(X)=1 are the spaces . Also, the only Asplund complex space X with absolute norm which satisfies n(2)(X)=1 is c0(Λ). 相似文献
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M. Anoussis 《Advances in Mathematics》2004,188(2):425-443
Let G be a compact group, not necessarily abelian, let ? be its unitary dual, and for f∈L1(G), let fn?f∗?∗f denote n-fold convolution of f with itself and f? the Fourier transform of f. In this paper, we derive the following spectral radius formula
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Summary If –I is a positive semidefinite operator andA andB are either both Hermitian or both unitary, then every unitarily invariant norm ofA–B is shown to be bounded by that ofA–B. Some related inequalities are proved. An application leads to a generalization of the Lidskii-Wielandt inequality to matrices similar to Hermitian.Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth 相似文献
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Fritz Gesztesy Mark Malamud Marius Mitrea Serguei Naboko 《Integral Equations and Operator Theory》2009,64(1):83-113
We study generalized polar decompositions of densely defined closed linear operators in Hilbert spaces and provide some applications
to relatively (form) bounded and relatively (form) compact perturbations of self-adjoint, normal, and m-sectorial operators.
Based upon work partially supported by the US National Science Foundation under Grant Nos. DMS-0400639 and FRG-0456306, and
the Austrian Science Fund (FWF) under Grant No. Y330. 相似文献
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Valeri Obukhovskii Pietro Zecca 《Nonlinear Analysis: Theory, Methods & Applications》2011,74(8):2765-2777
We introduce the notion of a multivalued causal operator and consider an abstract Cauchy problem in a Banach space for various classes of functional inclusions with causal operators. The methods of the topological degree theory for condensing maps are applied to obtain local and global existence results for this problem and to study the continuous dependence of a solution set on initial data. As application we generalize some existence results for semilinear functional differential inclusions and Volterra integro-differential inclusions with delay. 相似文献
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Michel Talagrand 《Publications Mathématiques de L'IHéS》1995,81(1):73-205
The concentration of measure phenomenon in product spaces roughly states that, if a set A in a product ΩN of probability spaces has measure at least one half, “most” of the points of Ωn are “close” to A. We proceed to a systematic exploration of this phenomenon. The meaning of the word “most” is made rigorous
by isoperimetrictype inequalities that bound the measure of the exceptional sets. The meaning of the work “close” is defined
in three main ways, each of them giving rise to related, but different inequalities. The inequalities are all proved through
a common scheme of proof. Remarkably, this simple approach not only yields qualitatively optimal results, but, in many cases,
captures near optimal numerical constants. A large number of applications are given, in particular to Percolation, Geometric
Probability, Probability in Banach Spaces, to demonstrate in concrete situations the extremely wide range of application of
the abstract tools.
Dedicated to Vitali Milman 相似文献
17.
Plamen Iliev 《Selecta Mathematica, New Series》2007,13(3):497-530
We consider the heat equation u
t
= Lu where L is a second-order difference operator in a discrete variable n. The fundamental solution has an expansion in terms of the Bessel functions of imaginary argument. The coefficients α
k
(n, m) in this expansion are analogs of Hadamard’s coefficients for the (continuous) Schr?dinger operator.
We derive an explicit formula for α
k
in terms of the wave and the adjoint wave functions of the Toda lattice hierarchy. As a first application of this result,
we prove that the values of these coefficients on the diagonals n = m and n = m + 1 define a hierarchy of differential-difference equations which is equivalent to the Toda lattice hierarchy. Using this
fact and the correspondence between commutative rings of difference operators and algebraic curves we show that the fundamental
solution can be summed up, giving a finite formula involving only two Bessel functions with polynomial coefficients in the
time variable t, if and only if the operator L belongs to the family of bispectral operators constructed in [18].
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The classical Krein-Naimark formula establishes a one-to-one correspondence between the generalized resolvents of a closed
symmetric operator in a Hilbert space and the class of Nevanlinna families in a parameter space. Recently it was shown by
V.A. Derkach, S. Hassi, M.M. Malamud and H.S.V. de Snoo that these parameter families can be interpreted as so-called Weyl
families of boundary relations, and a new proof of the Krein-Naimark formula in the Hilbert space setting was given with the
help of a coupling method. The main objective of this paper is to adapt the notion of boundary relations and their Weyl families
to the Krein space case and to prove some variants of the Krein-Naimark formula in an indefinite setting.
相似文献