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# Dissipative operator

In mathematics, a dissipative operator is a linear operator A defined on a dense linear subspace D(A) of a real or complex Hilbert space (H, ⟨ , ⟩), taking values in H, for which

$mathrm\left\{Re\right\} langle x, A x rangle leq 0$

for every x in D(A). If there exists a constant ω ≥ 0 such that

$mathrm\left\{Re\right\} langle x, A x rangle leq omega langle x, x rangle equiv omega | x |^\left\{2\right\}$

for every x in D(A), then A is said to be a quasidisspative operator.

## Examples

$x cdot A x = x cdot \left(-x\right) = - | x |^\left\{2\right\} leq 0,$

so A is a dissipative (and hence quasidissipative) operator.

• Consider H = L2([0, 1]; R) with its usual inner product, and let Au = u′ with domain D(A) equal to those functions u in the Sobolev space H1([0, 1]; R) with u(1) = 0. D(A) is dense in L2([0, 1]; R). Moreover, for every u in D(A), using integration by parts,

$langle u, A u rangle = int_\left\{0\right\}^\left\{1\right\} u\left(x\right) u\text{'}\left(x\right) , mathrm\left\{d\right\} x = - frac1\left\{2\right\} u\left(0\right)^\left\{2\right\} leq 0.$

Hence, A is a dissipative (and hence quasidissipative) operator.

$langle u, Delta u rangle = int_\left\{Omega\right\} u\left(x\right) Delta u\left(x\right) , mathrm\left\{d\right\} x = - int_\left\{Omega\right\} big| nabla u\left(x\right) big|^\left\{2\right\} , mathrm\left\{d\right\} x = - | nabla u |_\left\{L^\left\{2\right\} \left(Omega; mathbf\left\{R\right\}\right)\right\} leq 0,$

so the Laplacian is a dissipative (and hence quasidissipative) operator.

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