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Approximation of Semigroups and Related Operator Functions by Resolvent Series

Volker Grimm, Martin Gugat
2010 SIAM Journal on Numerical Analysis  
We consider the approximation of semigroups e τ A and of the functions ϕ j (τ A) that appear in exponential integrators by resolvent series.  ...  The interesting fact is that the resolvent series expresses the operator functions e τ A and ϕ j (τ A), respectively, in efficiently computable terms.  ...  Introduction and main results. In this note we will discuss approximations to semigroups and to the operator functions that appear in exponential Runge-Kutta methods by resolvent series.  ... 
doi:10.1137/090768084 fatcat:5o2neqihs5e3xhd6yakpnwkrfi

The fundamental properties of quasi-semigroups

Sutrima, Ch. Rini Indrati, Lina Aryati, Mardiyana
2017 Journal of Physics, Conference Series  
In this situation we can use the quasi semigroups theory as development of the two parameters semigroups. This semigroups is induced by bounded evolution operators ( ) that satisfy some assumptions.  ...  In this paper we determine the fundamental properties of the quasi semigroups included its generator related to the time-dependent evolution equation.  ...  Next, we define the resolvent operator of ( ) similar to one in operator theory. For each we define the resolvent operator of ( ) as ( ( )) ( ( )) , with its resolvent set ( ( )).  ... 
doi:10.1088/1742-6596/855/1/012052 fatcat:mocxkzg7krfsdghjgrkaih7ioi

Hagen Neidhardt (1950–2019) – His Work and Legacy

Jussi Behrndt, Pavel Exner, Takashi Ichinose, Mark Malamud, Valentin Zagrebnov
2020 EMS Newsletter  
The function (·) is called the spectral shift function of the pair {H, H 0 } and the relation itself, the trace formula.  ...  The function ξ(·) is called the spectral shift function of the pair {H, H 0 } and the relation itself, the trace formula.  ...  He had the knack of being able to reverse the mood in "unpleasant" situations by his good, amusing comments, but he could also become emotional and philosophical.  ... 
doi:10.4171/news/115/7 fatcat:dzb5zwgigfaj3cl6ioe6xa4rda

Approximate solutions of problems involving normal operators

W Lamb, J Mika, G.F Roach
1987 Journal of Mathematical Analysis and Applications  
In particular, polynomial approximations are obtained for resolvents and semigroups in terms of inverses and resolvents, respectively. 7"  ...  The spectral theory for unbounded normal operators is used to develop a systematic method of approximating functions of operators with other, more easily computable functions, leading to a priori error  ...  In particular we use the method for approximating semigroups generated by unbounded operators in terms of resolvents.  ... 
doi:10.1016/0022-247x(87)90086-2 fatcat:yjky2vayojfw5chdnj4w6viayy

A note on approximation of operator semigroups [article]

Jochen Glück
2016 arXiv   pre-print
We show that the operator semigroup (e^t(A-kP))_t > 0 converges to a semigroup on a subspace of X as k →∞ and we compute the limit semigroup.  ...  Let A be a bounded linear operator and P a bounded linear projection on a Banach space X.  ...  The spectrum of an operator A ∈ L(X) is denoted by σ(A) and for every λ ∈ C \ σ(A), R(λ, A) := (λ − A) −1 denotes the resolvent of A at λ.  ... 
arXiv:1511.02329v2 fatcat:rorimsmdznhexdqwyp7zvqxwuu

C_0-semigroups and resolvent operators approximated by Laguerre expansions [article]

Luciano Abadias, Pedro J. Miana
2014 arXiv   pre-print
We apply this result to approximate C_0-semigroups and resolvent operators in abstract Banach spaces.  ...  We study certain Laguerre functions, its Laplace transforms and the convergence of Laguerre series in Lebesgue spaces.  ...  Rezola and L. Roncal for the pieces of advice and assistance provided in order to obtain some previous results.  ... 
arXiv:1311.7542v2 fatcat:5btzcj52tnbbjllut4zdhargku

C0-semigroups and resolvent operators approximated by Laguerre expansions

Luciano Abadias, Pedro J. Miana
2017 Journal of Approximation Theory  
In this paper we introduce Laguerre expansions to approximate vector-valued functions. We apply this result to approximate C 0 -semigroups and resolvent operators in abstract Banach spaces.  ...  Finally, we illustrate the main results of this paper with some examples: shift, convolution and holomorphic semigroups, where the rate of convergence is improved. c  ...  Authors have been partially supported by Project MTM2013-42105-P and MTM2016-77710-P, DGI-FEDER, of the MCYTS; Project E-64, D.G. Aragón; and Project UZCUD2014-CIE-09, Universidad de Zaragoza.  ... 
doi:10.1016/j.jat.2016.09.001 fatcat:o7wragsaknempozi2g4cqcvy6m

Page 7659 of Mathematical Reviews Vol. , Issue 99k [page]

1999 Mathematical Reviews  
Then, from the relation between the resolvents of A and B, it is obtained that for every f € 2 the convergence rates of the ergodic limits of 74 and C, at infinity are the same as the convergence rate  ...  Then a formula relating the infinitesimal generator of S to the operators L and M is proven.  ... 

Page 1052 of Mathematical Reviews Vol. , Issue 99b [page]

1991 Mathematical Reviews  
The second ap- proach is based upon the analysis of the resolvent operator of positive semigroups. By using a result of L. W. Weis [Proc. Amer. Math.  ...  The second group of results is based on a relation between positive linear functionals majorized by a quasi-weight and ele- ments of so-called trio-commutants.  ... 

Automatic smoothness detection of the resolvent Krylov subspace method for the approximation of C_0-semigroups [article]

Volker Grimm, Tanja Göckler
2017 arXiv   pre-print
The resolvent Krylov subspace method builds approximations to operator functions f(A) times a vector v.  ...  For the semigroup and related operator functions, this method is proved to possess the favorable property that the convergence is automatically faster when the vector v is smoother.  ...  This work has been supported by the Deutsche Forschungsgemeinschaft (DFG) via GR 3787/1-1.  ... 
arXiv:1701.08046v1 fatcat:xnjeton3j5bbtpwenoqzybhrom

Automatic Smoothness Detection of the Resolvent Krylov Subspace Method for the Approximation of $C_0$-Semigroups

Volker Grimm, Tanja Göckler
2017 SIAM Journal on Numerical Analysis  
The resolvent Krylov subspace method builds approximations to operator functions f (A) times a vector v.  ...  For the semigroup and related operator functions, this method is proved to possess the favorable property that the convergence is automatically faster when the vector v is smoother.  ...  This work has been supported by the Deutsche Forschungsgemeinschaft (DFG) via GR 3787/1-1.  ... 
doi:10.1137/15m104880x fatcat:w4q7bnlun5h3vli2uphyznemuu

On the theory of semigroups of operators on locally convex spaces

Benjamin Dembart
1974 Journal of Functional Analysis  
The action of % is characterized by convolution against the semigroup, and the semigroup is computed as the limit of '% acting on an approximate identity.  ...  A space X of continuous E-valued functions is defined for a locally convex space E, and the generalized resolvent % of an operator A on E is defined as an operator on X.  ...  Applying Lemma 2, we see 'iR&' *f = bn' * %f E D(B) and %%+,' t f = 'B&' * %f = 4,' * S'iRf. However, B%f E '9J, so we may apply this  ... 
doi:10.1016/0022-1236(74)90061-5 fatcat:33zkv6mv3fhohaustgaa2kuc4y

Tensor products of closed operators on Banach spaces

Michael Reed, Barry Simon
1973 Journal of Functional Analysis  
Using the Gelfand theory and resolvent algebra techniques, a spectral mapping theorem is proven for a certain class of rational functions of A and B.  ...  Let A and B be closed operators on Banach spaces X and Y. Assume that A and B have nonempty resolvent sets and that the spectra of A and B are unbounded. Let 01 be a uniform cross norm on X @ Y.  ...  ACKNOWLEDGMENT The authors would like to thank the University of Michigan where this manuscript was prepared while one of the authors (MR.) held a visiting appointment.  ... 
doi:10.1016/0022-1236(73)90038-4 fatcat:ubs5j5f76zbknp2xj44s5luxya

Nonlinear Principal Components and Long-run Implications of Multivariate Diffusions [article]

Xioahong Chen, Lars Peter Hansen, Jose Scheinkman
2009 arXiv   pre-print
By exploiting the theory of continuous-time, reversible Markov diffusion processes, we give a different interpretation of these NPCs and the smoothness constraints.  ...  of the overidentifying restrictions implied by such a process from low frequency data.  ...  We may invert this latter relation and obtain: (5.5) G α φ = ∞ 0 exp(−αt) exp(−tF )φdt which is the usual formula for the resolvents of a semigroup of operators.  ... 
arXiv:0908.0547v1 fatcat:vh6llqvzfzbtfm7u2lqlkbzqyq

Nonlinear principal components and long-run implications of multivariate diffusions

Xiaohong Chen, Lars Peter Hansen, José Scheinkman
2009 Annals of Statistics  
By exploiting the theory of continuous-time, reversible Markov diffusion processes, we give a different interpretation of these NPCs and the smoothness constraints.  ...  of the overidentifying restrictions implied by such a process from low-frequency data.  ...  Brian Davies, Nan Li, Oliver Linton, Peter McCullagh, Nour Meddahi, Gigliola Staffilani, Stephen Stigler, Gauhar Turmuhambetova and Noah Williams for useful conversations.  ... 
doi:10.1214/09-aos706 fatcat:oe2r7xgzcfcvhjz7jqh6okmd4a
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