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

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

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

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

Examples

• For a simple finite-dimensional example, consider n-dimensional Euclidean space Rⁿ with its usual dot product. If A denotes the negative of the identity operator, defined on all of Rⁿ, then

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

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,

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

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

• Consider H = H₀²(Ω; R) for an open and connected domain Ω ⊆ Rⁿ and let A = Δ, the Laplace operator, defined on the dense subspace of compactly supported smooth functions on Ω. Then, using integration by parts,

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

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

References

• Renardy, Michael and Rogers, Robert C. (2004). An introduction to partial differential equations. Second edition, New York: Springer-Verlag. ISBN 0-387-00444-0. (Definition 11.25)