35 research outputs found
Boundary Conditions for Singular Perturbations of Self-Adjoint Operators
Let A:D(A)\subseteq\H\to\H be an injective self-adjoint operator and let
\tau:D(A)\to\X, X a Banach space, be a surjective linear map such that
\|\tau\phi\|_\X\le c \|A\phi\|_\H. Supposing that \text{\rm Range}
(\tau')\cap\H' =\{0\}, we define a family of self-adjoint
operators which are extensions of the symmetric operator .
Any in the operator domain is characterized by a sort
of boundary conditions on its univocally defined regular component \phireg,
which belongs to the completion of D(A) w.r.t. the norm \|A\phi\|_\H. These
boundary conditions are written in terms of the map , playing the role of
a trace (restriction) operator, as \tau\phireg=\Theta Q_\phi, the extension
parameter being a self-adjoint operator from X' to X. The self-adjoint
extension is then simply defined by A^\tau_\Theta\phi:=A \phireg. The case in
which is a convolution operator on LD, T a distribution with
compact support, is studied in detail.Comment: Revised version. To appear in Operator Theory: Advances and
Applications, vol. 13
Asymptotics of solutions of the heat equation in cones and dihedra under minimal assumptions on the boundary
Besov spaces with variable smoothness and integrability on Lie groups of polynomial growth
Mixed norm variable exponent Bergman space on the unit disc
We introduce and study the mixed norm variable order Bergman space A(q,p(.)) (D), 1 1 from inside the interval I = (0, 1). The situation is quite different in the cases p(1) 1, when A(2,p(.)) (D) = H-2(D) isometrically, and when this is not longer true