Knowlpedia

Bounded Operator on a

A whose action does not increase vector norms by more than a fixed constant; equivalently, a continuous linear map.

Let HH be a (complex) Hilbert space. A linear operator A:H→HA:H\to H is bounded if there exists a constant C≥0C\ge 0 such that

∥Ax∥≤C∥x∥for all x∈H.\|Ax\|\le C\|x\| \quad \text{for all } x\in H.

The smallest such constant is the

∥A∥:=sup⁡∥x∥=1∥Ax∥.\|A\| := \sup_{\|x\|=1}\|Ax\|.
Equivalent characterizations

For linear maps between (in particular, ), the following are equivalent:

  • AA is bounded.
  • AA is continuous (everywhere).
  • AA is continuous at 00.

In finite dimension (e.g. on a ), every is bounded.

Adjoint and basic properties

If AA is bounded on a Hilbert space, then there exists a unique bounded adjoint A∗A^\ast such that

⟨x,Ay⟩=⟨A∗x,y⟩for all x,y∈H.\langle x,Ay\rangle = \langle A^\ast x,y\rangle \quad \text{for all } x,y\in H.

Key norm identities/inequalities:

  • ∥A∗∥=∥A∥\|A^\ast\|=\|A\|.
  • ∥AB∥≤∥A∥ ∥B∥\|AB\|\le \|A\|\,\|B\|.
  • ∥A∥2=∥A∗A∥\|A\|^2 = \|A^\ast A\|.

An operator AA is when A=A∗A=A^\ast.

Example (matrices)

On H=CnH=\mathbb{C}^n, every matrix A∈Cn×nA\in \mathbb{C}^{n\times n} defines a bounded operator, and ∥A∥\|A\| is the induced norm

∥A∥=sup⁡x≠0∥Ax∥2∥x∥2,\|A\|=\sup_{x\ne 0}\frac{\|Ax\|_2}{\|x\|_2},

equal to the largest singular value of AA.