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18
The Theory Of Generalized Dirichlet Forms And Its Applications In Analysis And Stochastics
, 1996
"... We present an introduction (also for nonexperts) to a new framework for the analysis of (up to) second order differential operators (with merely measurable coefficients and in possibly infinitely many variables) on L²spaces via associated bilinear forms. This new framework, in particular, covers b ..."
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We present an introduction (also for nonexperts) to a new framework for the analysis of (up to) second order differential operators (with merely measurable coefficients and in possibly infinitely many variables) on L²spaces via associated bilinear forms. This new framework, in particular, covers both the elliptic and the parabolic case within one approach. To this end we introduce a new class of bilinear forms, socalled generalized Dirichlet forms, which are in general neither symmetric nor coercive, but still generate associated C0 semigroups. Particular examples of generalized Dirichlet forms are symmetric and coercive Dirichlet forms (cf. [FOT], [MR1]) as well as time dependent Dirichlet forms (cf. [O1]). We discuss many applications to differential operators that can be treated within the new framework only, e.g. parabolic differential operators with unbounded drifts satisfying no L p conditions, singular and fractional diffusion operators. Subsequently, we analyz...
Dirac Structures and Boundary Control Systems Associated with SkewSymmetric Differential Operators, Memorandum Faculteit TW 1730, Universiteit Twente
"... Abstract. Associated with a skewsymmetric linear operator on the spatial domain [a, b] we define a Dirac structure which includes the port variables on the boundary of this spatial domain. This Dirac structure is a subspace of a Hilbert space. Naturally, associated with this Dirac structure is an i ..."
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Cited by 10 (7 self)
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Abstract. Associated with a skewsymmetric linear operator on the spatial domain [a, b] we define a Dirac structure which includes the port variables on the boundary of this spatial domain. This Dirac structure is a subspace of a Hilbert space. Naturally, associated with this Dirac structure is an infinitedimensional system. We parameterize the boundary port variables for which the C0semigroup associated with this system is contractive or unitary. Furthermore, this parameterization is used to split the boundary port variables into inputs and outputs. Similarly, we define a linear port controlled Hamiltonian system associated with the previously defined Dirac structure and a symmetric positive operator defining the energy of the system. We illustrate this theory on the example of the Timoshenko beam.
First Order Perturbations Of Dirichlet Operators: Existence And Uniqueness
, 1996
"... We study perturbations of type B \Delta r of Dirichlet operators (L 0 ; D(L 0 )) associated with Dirichlet forms of type E 0 (u; v) = 1=2 R hru; rviH d¯ on L 2 (E; ¯) where E is a finite or infinite dimensional Banach space E. Here H denotes a Hilbert space densely and continuously embed ..."
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Cited by 5 (1 self)
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We study perturbations of type B \Delta r of Dirichlet operators (L 0 ; D(L 0 )) associated with Dirichlet forms of type E 0 (u; v) = 1=2 R hru; rviH d¯ on L 2 (E; ¯) where E is a finite or infinite dimensional Banach space E. Here H denotes a Hilbert space densely and continuously embedded in E. Assuming quasiregularity of (E 0 ; D(E 0 )) we show that there always exists a closed extension of Lu := L 0 u + hB; ruiH that generates a subMarkovian C 0 semigroup of contractions on L 2 (E; ¯) (resp. L 1 (E; ¯)), if B 2 L 2 (E; H;¯) and R hB; ruiH d¯ 0; u 0. If D is an appropriate core for (L 0 ; D(L 0 )) we show that there is only one closed extension of (L; D) in L 1 (E; ¯) generating a strongly continuous semigroup. In particular we apply our results to operators of type \Delta H +B \Delta r, where \Delta H denotes the GrossLaplacian on an abstract Wiener space (E; H; fl) and B = \Gammaid E + v, where v takes values in the CameronMartin s...
Boundary relations and their Weyl families
 Transactions of the American Mathematical Society
, 2006
"... Abstract. The concepts of boundary relations and the corresponding Weyl families are introduced. Let S be a closed symmetric linear operator or, more generally, a closed symmetric relation in a Hilbert space H, letH be an auxiliary Hilbert space, let ..."
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Abstract. The concepts of boundary relations and the corresponding Weyl families are introduced. Let S be a closed symmetric linear operator or, more generally, a closed symmetric relation in a Hilbert space H, letH be an auxiliary Hilbert space, let
Contraction semigroups on metric graphs
 Analysis on Graphs and its Applications, volume 77 of Proceedings of Symposia in Pure Mathematics
, 2008
"... Dedicated to Volker Enss on the occasion of his 65th birthday ..."
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Dedicated to Volker Enss on the occasion of his 65th birthday
ABSTRACT WAVE EQUATIONS AND ASSOCIATED DIRACTYPE OPERATORS
, 2010
"... We discuss the unitary equivalence of generators GA,R associated with abstract damped wave equations of the type ü+R ˙u+A ∗Au = 0 in some Hilbert space H1 and certain nonselfadjoint Diractype operators QA,R (away from the nullspace of the latter) in H1 ⊕ H2. The operator QA,R represents a nons ..."
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We discuss the unitary equivalence of generators GA,R associated with abstract damped wave equations of the type ü+R ˙u+A ∗Au = 0 in some Hilbert space H1 and certain nonselfadjoint Diractype operators QA,R (away from the nullspace of the latter) in H1 ⊕ H2. The operator QA,R represents a nonselfadjoint perturbation of a supersymmetric selfadjoint Diractype operator. Special emphasis is devoted to the case where 0 belongs to the continuous spectrum of A∗A. In addition to the unitary equivalence results concerning GA,R and QA,R, we provide a detailed study of the domain of the generator GA,R, consider spectral properties of the underlying quadratic operator pencil M(z) = A  2 − izR − z2IH1, z ∈ C, derive a family of conserved quantities for abstract wave equations in the absence of damping, and prove equipartition of energy for supersymmetric selfadjoint Diractype operators. The special example where R represents an appropriate function of A  is treated in depth and the semigroup growth bound for this example is explicitly computed and shown to coincide with the corresponding spectral bound for the underlying generator and also with that of the corresponding Diractype operator. The cases of undamped (R = 0) and damped (R ̸ = 0) abstract wave equations as well as the cases A∗A ≥ εIH1 for some ε> 0 and 0 ∈ σ(A∗A) (but 0 not an eigenvalue of A∗A) are separately studied in detail.
SELFADJOINT EXTENSIONS OF SYMMETRIC SUBSPACES
"... A theory of selfadjoint extensions of closed symmetric linear manifolds beyond the original space is presented. It is based on the Cayley transform of linear manifolds. Resolvent and spectral families of such extensions are characterized. These extensions are also determined by means of analytic c ..."
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A theory of selfadjoint extensions of closed symmetric linear manifolds beyond the original space is presented. It is based on the Cayley transform of linear manifolds. Resolvent and spectral families of such extensions are characterized. These extensions are also determined by means of analytic contractions between the "deficiency spaces " of the original symmetric linear manifold. 1 * Introduction * Let § be a Hubert space over the complex numbers C and denote by ξ>2 the Hubert space § © §. The adjoint Γ * of a linear manifold T in £>2 is a closed linear manifold defined by T * {{h, k] e &/(g, h) = (/, k) for all {/, g] e T}.
Conditions for Dissipative Symmetric Linear Differential Operators *
"... This paper is concerned with the linear differential operator m a j1 ax5 L = 2 A i + B, ..."
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This paper is concerned with the linear differential operator m a j1 ax5 L = 2 A i + B,
REGULARIZATION OF TWOTERM DIFFERENTIAL EQUATIONS WITH SINGULAR COEFFICIENTS BY QUASIDERIVATIVES
"... We propose a regularization of the formal differential expression l.y / D imy.m/.t/C q.t/y.t/; t 2.a; b/; of order m 3 by quasiderivatives. It is assumed that the distribution coefficient q has the antiderivative Q 2 L.Œa; bIC / : In the symmetric case.Q D Q/; we describe selfadjoint and maximal ..."
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We propose a regularization of the formal differential expression l.y / D imy.m/.t/C q.t/y.t/; t 2.a; b/; of order m 3 by quasiderivatives. It is assumed that the distribution coefficient q has the antiderivative Q 2 L.Œa; bIC / : In the symmetric case.Q D Q/; we describe selfadjoint and maximal dissipative/accumulative extensions of the minimal operator and its generalized resolvents. In the general (nonselfadjoint) case, we establish the conditions of convergence for the resolvents of the analyzed operators in norm. The case where m D 2 and Q 2 L2.Œa; bIC / was studied earlier. 1.
COMPONENTWISE AND CARTESIAN DECOMPOSITIONS OF LINEAR RELATIONS
, 906
"... Dedicated to Schôichi Ôta on the occasion of his sixtieth birthday Abstract. Let A be a, not necessarily closed, linear relation in a Hilbert space H with a multivalued part mul A. An operator B in H with ran B ⊥ mul A ∗∗ is said to be an operator part of A when A = B b+ ({0} × mul A), where the su ..."
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Dedicated to Schôichi Ôta on the occasion of his sixtieth birthday Abstract. Let A be a, not necessarily closed, linear relation in a Hilbert space H with a multivalued part mul A. An operator B in H with ran B ⊥ mul A ∗∗ is said to be an operator part of A when A = B b+ ({0} × mul A), where the sum is componentwise (i.e. span of the graphs). This decomposition provides a counterpart and an extension for the notion of closability of (unbounded) operators to the setting of linear relations. Existence and uniqueness criteria for the existence of an operator part are established via the socalled canonical decomposition of A. In addition, conditions are developed for the decomposition to be orthogonal (components defined in orthogonal subspaces of the underlying space). Such orthogonal decompositions are shown to be valid for several classes of relations. The relation A is said to have a Cartesian decomposition if A = U + i V, where U and V are symmetric relations and the sum is operatorwise. The connection between a Cartesian decomposition of A and