Finite-rank operator

In functional analysis, a branch of mathematics, a finite-rank operator is a bounded linear operator between Banach spaces whose range is finite-dimensional.

Finite-rank operators are matrices (of finite size) transplanted to the infinite dimensional setting. As such, these operators may be described via linear algebra techniques.

From linear algebra, we know that a rectangular matrix, with complex entries, ${\displaystyle M\in \mathbb {C} ^{n\times m}}$ has rank ${\displaystyle 1}$ if and only if ${\displaystyle M}$ is of the form

Exactly the same argument shows that an operator ${\displaystyle T}$ on a Hilbert space ${\displaystyle H}$ is of rank ${\displaystyle 1}$ if and only if

where the conditions on ${\displaystyle \alpha ,u,v}$ are the same as in the finite dimensional case.

Therefore, by induction, an operator ${\displaystyle T}$ of finite rank ${\displaystyle n}$ takes the form

where ${\displaystyle \{u_{i}\}}$ and ${\displaystyle \{v_{i}\}}$ are orthonormal bases. Notice this is essentially a restatement of singular value decomposition. This can be said to be a canonical form of finite-rank operators.

Generalizing slightly, if ${\displaystyle n}$ is now countably infinite and the sequence of positive numbers ${\displaystyle \{\alpha _{i}\}}$ accumulate only at ${\displaystyle 0}$, ${\displaystyle T}$ is then a compact operator, and one has the canonical form for compact operators.