Bounded Operators on Hilbert Space
A guided tour through the theory of bounded linear operators on a Hilbert space — from the geometry of the underlying space to norms, adjoints, spectra, and compactness.
Let $H$ be a Hilbert space over $\mathbb{C}$ (or $\mathbb{R}$). The set $B(H)$ of bounded linear operators $T : H \to H$ is itself a rich mathematical object: a Banach space under the operator norm, a $C^*$-algebra under composition and adjoint, and the natural home for spectral theory. This site walks through the core ideas, in order, with definitions, theorems, and worked examples at each stage.
Contents
Hilbert Spaces
Inner products, norms, completeness, orthogonality, and orthonormal bases.
02Bounded Linear Operators
Definition of boundedness, equivalence with continuity, and $B(H)$.
03The Operator Norm
Equivalent formulas for $\|T\|$, submultiplicativity, and completeness of $B(H)$.
04The Adjoint
Existence and uniqueness of $T^*$, its properties, and the $C^*$-identity.
05Special Classes
Self-adjoint, unitary, normal, and projection operators.
06Spectrum
Resolvent set, spectrum, spectral radius, and spectra of special operators.
07Compact Operators
Compact operators as limits of finite-rank operators, and their spectral theory.
08Worked Examples
Multiplication, shift, and integral operators, computed explicitly.