About 9,330 results
|If dom(A') = H and if A is symmetric, then from the closed graph theorem it follows |
that K is bounded and hence self-adjoint. So that a symmetric everywhere
defined operator is bounded and selfadjoint. A. 16. Essentially self-adjoint
operators A ...
|Show that B is a bounded linear operator on H. Show that B is self-adjoint and |
that A = B~l exists. (e) Show that ... Let L be a densely defined bounded
symmetric operator on a Hilbert space H. Show that L is essentially self-adjoint. 9.
Let M be ...
|(b) Show that if A is self-adjoint, then so is A\j. (c) Assume that A is closed. Show |
that Au and A have the same spectrum and the same eigenvalues. 2.22. Show
that a bounded symmetric operator is essentially self-adjoint. 2.23. (a) Let A be ...
|(A.2.1) For a symmetric operator A, its adjoint operator A∗ is defined on the |
domain D(A∗) consisting of the points x ∈ H for ... Except for bounded operators
(bounded symmetric operators are self-adjoint), there is a huge difference
between symmetric and self-adjoint operators. ... a unique self-adjoint extension
to some larger domain D(L), then L is said to be essentially self-adjoint (this
notion refers both ...
|Then S is essentially selfadjoint if and only if T is; in this case S and T have the |
same domain. In particular, S is selfadjoint if and only if T is. 2. The case of
relative bound 1 In the theorems proved in the preceding paragraph, the
|An operator a is said to be bounded on D(a) if there exists a real number r such |
that ||ae||<r||e||, e G £>(o). If otherwise, it ... For bounded operators, the notions of
symmetric, self-adjoint and essentially self-adjoint operators coincide. Let fibe a ...
|C8 A^ symmetric operator Ajs essentially selfadjoint if and only if its closure A^ is |
selfadjoint, i.e., A^ = A^. Given that ... Two bounded selfadjoint operators A\ and
An are said to commute if their commutator [AUA2] = A1A2-A2A1 (2.85) vanishes.
|Hence, C1IB(AIi,51)—l is a bounded operator with domain D(C) I H and norm |
less than 1. Therefore ... Proposition 8.6 Suppose that A is an essentially self-
adjoint operator and B is a symmetric operator on H such that D(B) Q D(A).
|But A* is densely defined and closed, by Theorem 1.5(2) we have A*** = A*, thus |
A* = A**, i.e., A* is self-adjoint. ... theorem gives a useful characterization for a
symmetric, bounded below operator to be self-adjoint or essentially self-adjoint.
|Then, At±A2 is essentially self-adjoint. Proof: The proof follows directly from the |
spectral theorem, y A useful criterion for essential self-adjointness is given in :
Lemma 3. Let A be symmetric operator and let (A+I)~l be bounded and densely ...