Dua Formula Eksponensial (Operator Linear dari Semigrup)

Assumed A is infinitesimal generator of C0-semigroup T(t) on X. This could be defined as T(t)=etA, applies if A is a bounded linear operator. Not if A is unbounded linear operator, then it will result in one possibility that show T(t) could be represented as etA. This paper will discuss and detail the proof of the other two formulas that show T(t) could be represented as etA.

Toeplitz Operators whose Symbols Are Borel Measures

In this paper, we are concerned with Toeplitz operators whose symbols are complex Borel measures. When a complex Borel measure μ on the unit circle is given, we give a formal definition of a Toeplitz operator T μ with symbol μ , as an unbounded linear operator on the Hardy space. We then study various properties of T μ . Among them, there is a theorem that the domain of T μ is represented by a trichotomy. Also, it was shown that if the domain of T μ contains at least one polynomial, then T μ is densely defined. In addition, we give evidence for the conjecture that T μ with a singular measure μ reduces to a trivial linear operator.

Krylov Solvability of Unbounded Inverse Linear Problems

The abstract issue of ‘Krylov solvability’ is extensively discussed for the inverse problem $$Af=g$$ A f = g where A is a (possibly unbounded) linear operator on an infinite-dimensional Hilbert space, and g is a datum in the range of A. The question consists of whether the solution f can be approximated in the Hilbert norm by finite linear combinations of $$g,Ag,A^2g,\dots $$ g , A g , A 2 g , ⋯ , and whether solutions of this sort exist and are unique. After revisiting the known picture when A is bounded, we study the general case of a densely defined and closed A. Intrinsic operator-theoretic mechanisms are identified that guarantee or prevent Krylov solvability, with new features arising due to the unboundedness. Such mechanisms are checked in the self-adjoint case, where Krylov solvability is also proved by conjugate-gradient-based techniques.

Approximate Controllability of Semilinear Impulsive Evolution Equations

We prove the approximate controllability of the following semilinear impulsive evolution equation:z'=Az+Bu(t)+F(t,z,u), z∈Z, t∈(0,τ], z(0)=z0, z(tk+)=z(tk-)+Ik(tk,z(tk),u(tk)), k=1,2,3,…,p,where0<t1<t2<t3<⋯<tp<τ,ZandUare Hilbert spaces,u∈L2(0,τ;U),B:U→Zis a bounded linear operator,Ik,F:[0,τ]×Z×U→Zare smooth functions, andA:D(A)⊂Z→Zis an unbounded linear operator inZwhich generates a strongly continuous semigroup{T(t)}t≥0⊂Z. We suppose thatFis bounded and the linear system is approximately controllable on[0,δ]for allδ∈(0,τ). Under these conditions, we prove the following statement: the semilinear impulsive evolution equation is approximately controllable on[0,τ].

Perron-Frobenius and Krein-Rutman theorems for tangentially positive operators

We study several aspects of a generalized Perron-Frobenius and Krein-Rutman theorems concerning spectral properties of a (possibly unbounded) linear operator on a cone in a Banach space. The operator is subject to the so-called tangency or weak range assumptions implying the resolvent invariance of the cone. The further assumptions rely on relations between the spectral and essential spectral bounds of the operator. In general we do not assume that the cone induces the Banach lattice structure into the underlying space.

Improved Regularization Method for Backward Cauchy Problems Associated with Continuous Spectrum Operator

We consider in this paper an abstract parabolic backward Cauchy problem associated with an unbounded linear operator in a Hilbert space , where the coefficient operator in the equation is an unbounded self-adjoint positive operator which has a continuous spectrum and the data is given at the final time and a solution for is sought. It is well known that this problem is illposed in the sense that the solution (if it exists) does not depend continuously on the given data. The method of regularization used here consists of perturbing both the equation and the final condition to obtain an approximate nonlocal problem depending on two small parameters. We give some estimates for the solution of the regularized problem, and we also show that the modified problem is stable and its solution is an approximation of the exact solution of the original problem. Finally, some other convergence results including some explicit convergence rates are also provided.

Interior Controllability of a Broad Class of Reaction Diffusion Equations

We prove the interior approximate controllability of the following broad class of reaction diffusion equation in the Hilbert spacesZ=L2(Ω)given byz′=−Az+1ωu(t),t∈[0,τ], whereΩis a domain inℝn,ωis an open nonempty subset ofΩ,1ωdenotes the characteristic function of the setω, the distributed controlu∈L2(0,t1;L2(Ω))andA:D(A)⊂Z→Zis an unbounded linear operator with the following spectral decomposition:Az=∑j=1∞λj∑k=1γj〈z,ϕj,k〉ϕj,k. The eigenvalues0<λ1<λ2<⋯<⋯λn→∞ofAhave finite multiplicityγjequal to the dimension of the corresponding eigenspace, and{ϕj,k}is a complete orthonormal set of eigenvectors ofA. The operator−Agenerates a strongly continuous semigroup{T(t)}given byT(t)z=∑j=1∞e−λjt∑k=1γj〈z,ϕj,k〉ϕj,k. Our result can be applied to thenD heat equation, the Ornstein-Uhlenbeck equation, the Laguerre equation, and the Jacobi equation.

A Modified Quasi-Reversibility Method for a Class of Ill-Posed Cauchy Problems

The goal of this paper is to present some extensions of the method of quasi-reversibility applied to an ill-posed Cauchy problem associated with an unbounded linear operator in a Hilbert space. The key point to our proof is the use of a new perturbation to construct a family of regularizing operators for the considered problem. We show the convergence of this method, and we estimate the convergence rate under a priori regularity assumptions on the problem data.