Bounded Operator on a Hilbert Space

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

The smallest such constant is the operator norm

Bounded operators are the standard class of operators used to model physical transformations and observables in many finite-dimensional quantum settings.

Equivalent characterizations

For linear maps between normed vector spaces (in particular, Hilbert spaces), the following are equivalent:

• $A$ is bounded.
• $A$ is continuous (everywhere).
• $A$ is continuous at $0$.

In finite dimension (e.g. on a complex Hilbert space ), every linear operator is bounded.

Adjoint and basic properties

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

Key norm identities/inequalities:

• $\|A^\ast\|=\|A\|$.
• $\|AB\|\le \|A\|\,\|B\|$.
• $\|A\|^2 = \|A^\ast A\|$.

An operator $A$ is self-adjoint when $A=A^\ast$.

Example (matrices)

On $H=\mathbb{C}^n$, every matrix $A\in \mathbb{C}^{n\times n}$ defines a bounded operator, and $\|A\|$ is the induced norm

equal to the largest singular value of $A$.