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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...
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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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...
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
"... Abstract. 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 ..."
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Abstract. 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. 1.
THE EXISTENCE PROBLEM FOR DYNAMICS OF DISSIPATIVE SYSTEMS IN QUANTUM PROBABILITY
, 2002
"... We consider hermitian dissipative mappings δ which are densely defined in unital C ∗algebras A. The identity element in A is also in the domain of δ. Completely dissipative maps δ are defined by the requirement that the induced maps, (aij) → (δ(aij)), are dissipative on the n by n complex matrices ..."
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We consider hermitian dissipative mappings δ which are densely defined in unital C ∗algebras A. The identity element in A is also in the domain of δ. Completely dissipative maps δ are defined by the requirement that the induced maps, (aij) → (δ(aij)), are dissipative on the n by n complex matrices over A for all n. We establish the existence of different types of maximal extensions of completely dissipative maps. If the enveloping von Neumann algebra of A is injective, we show the existence of an extension of δ which is the infinitesimal generator of a quantum dynamical semigroup of completely positive maps in the von Neumann algebra. If δ is a given wellbehaved ∗derivation, then we show that each of the maps ±δ is completely dissipative. 1.
OPERATOR HOLES AND EXTENSIONS OF SECTORIAL OPERATORS AND DUAL PAIRS OF CONTRACTIONS
, 2005
"... Abstract. A description of the set of msectorial extensions of a dual pair {A1, A2} of nonnegative operators is obtained. Some classes of nonaccretive extensions of the dual pair {A1, A2} are described too. Both problems are reduced to similar problems for a dual pair {T1, T2} of nondensely defined ..."
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Abstract. A description of the set of msectorial extensions of a dual pair {A1, A2} of nonnegative operators is obtained. Some classes of nonaccretive extensions of the dual pair {A1, A2} are described too. Both problems are reduced to similar problems for a dual pair {T1, T2} of nondensely defined symmetric contractions Tj = (I−Aj)(I+Aj) −1, j ∈ {1, 2}. In turn these problems are reduced to the investigation of the corresponding operator ”holes”. A complete description of the set of all proper and improper extensions of a nonnegative operator is obtained too. 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