Finite rank, relatively bounded perturbations of semigroup generators. (English) Zbl 0602.47029
Let A be the (linear) infinitesimal generator of a strongly-continuous semigroup (or group) of operators on the Hilbert space Y and let P be a (generally unbounded) linear operator in Y, assumed to be A-bounded and of one dimensional range. P is, typically, nondissipative. Does \(A+P\) generate a strongly-continuous semigroup on Y? On the negative side, the following result is shown (by means of constructive counter-examples): Let A be a unitary group generator and let P be \(A^{\epsilon}\)-bounded, \(\epsilon >0\) arbitrary (in addition to P having one dimensional range). Yet \(A+P\) is not a generator of a strongly continuous semigroup on Y.
On the positive side, a sufficient condition is given for generation of \(A+P\). This sufficient condition is, in particular, satisfied by certain hyperbolic equations in boundary feedback closed loop form: this is due to ”sharp” trace theory results for second order hyperbolic equations recently established by the authors.
On the positive side, a sufficient condition is given for generation of \(A+P\). This sufficient condition is, in particular, satisfied by certain hyperbolic equations in boundary feedback closed loop form: this is due to ”sharp” trace theory results for second order hyperbolic equations recently established by the authors.
Keywords:
infinitesimal generator; strongly-continuous semigroup; hyperbolic equations in boundary feedback closed loop formReferences:
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