nLab
self-adjoint operator

Contents

Definition

Let A:HHA: H\to H be an unbounded operator on a Hilbert space HH. An unbounded operator A *A^* is its adjoint? if

  • (Axy)=(xA *y)(Ax|y) = (x|A^*y) for all xdom(A)x\in dom(A) and ydom(A *)y\in dom(A^*); and
  • every BB satisfying the above property for A *A^* is a restriction of AA.

An adjoint does not need to exist in general.

An unbounded operator is symmetric if dom(A)dom(A *)dom(A)\subset dom(A^*) and Ax=A *xA x = A^*x for all xdom(A)x\in dom(A) (one also writes AA *A\subset A^*).

The domain of A *A^* is the set of all vectors yHy\in H such that the linear functional x(Axy)x\mapsto (Ax|y) is bounded on dom(A)dom(A).

The graph Γ AHH\Gamma_A\subset H\oplus H satisfies Γ A *=τ(Γ A) \Gamma_{A^*} = \tau(\Gamma_A)^\perp where \perp denotes the orthogonal complement and τ\tau denotes the transposition of the direct summands changing the sign of one of the factors, i.e. xyyxx\oplus y\mapsto -y\oplus x. An unbounded operator AA is closed if Γ A\Gamma_A is closed subspace of HHH\oplus H. An operator BB is a closure of an operator AA if Γ B\Gamma_B is a closure of operator Γ A\Gamma_A. It is said that BB is an extension of AA and one writes BAB\supset A if Γ BΓ A\Gamma_B\supset \Gamma_A. The closure of an unbounded operator does not need to exist.

For any unbounded operator AA with a dense dom(A)Hdom(A)\subset H, if the adjoint operator A *A^* exists, then A *A^* is closed, and if (A *) *(A^*)^* exists then it coincides with a closure of AA.

An unbounded operator A:HHA : H\to H on a Hilbert space HH is self-adjoint if

  • it has a densely defined domain dom(A)Hdom(A)\subset H
  • A=A *A = A^*, i.e. dom(A *)=dom(A)dom(A^*)= dom(A) and Ax=A *xA x = A^* x for all xdom(A)x\in dom(A)

An (unbounded) operator is essentially self-adjoint if it is symmetric and its spectrum (as a subspace of the complex plane) is contained in the real line. Alternatively, it is symmetric if its closure is self-adjoint.

A Hermitean (or hermitian) operator is a bounded symmetric operator (which is necessarily self-adjoint), although some authors use the term for any self-adjoint operator.

For a bounded operator A:HKA: H\to K between Hilbert spaces, define the Hermitean conjugate operator A *:KHA^*: K\to H by (Axy) H=(xA *y) K(Ax|y)_H = (x|A^*y)_K, for all xKx\in K, yHy\in H. Distinguish it from the concept of the transposed operator? A T:K *H *A^T: K^*\to H^* between the dual spaces.

In an arbitrary **-algebra, a self-adjoint or hermitian element is any element AA such that A *=AA^* = A.

References

  • A. A. Kirillov, A. D. Gvišiani, Теоремы и задачи функционального анализа (theorems and exercises in functional analysis), Moskva, Nauka 1979, 1988

  • A. B. Antonevič, Ja. B. Radyno, Funkcional’nij analiz i integral’nye uravnenija, Minsk 1984

  • S. Kurepa, Funkcionalna analiza, elementi teorije operatora, Školska knjiga, Zagreb 1981.

  • Reed, M.; Simon, B.: Methods of modern mathematical physics. Volume 1, Functional Analysis

  • Walter Rudin, Functional analysis

Revised on October 27, 2013 08:49:50 by Toby Bartels (98.19.40.208)