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 on a Hilbert space

A canonical form

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, ''M'' ∈ C^{''n'' × ''m''} has rank 1 if and only if ''M'' is of the form

:$M = \alpha \cdot u v^*, \quad \mbox \quad \, u \, = \, v\, = 1 \quad \mbox \quad \alpha \geq 0 .$

Exactly the same argument shows that an operator ''T'' on a Hilbert space ''H'' is of rank 1 if and only if

:$T h = \alpha \langle h, v\rangle u \quad \mbox \quad h \in H ,$

where the conditions on ''α'', ''u'', and ''v'' are the same as in the finite dimensional case.

Therefore, by induction, an operator ''T'' of finite rank ''n'' takes the form

:$T h = \sum _ ^n \alpha_i \langle h, v_i\rangle u_i \quad \mbox \quad h \in H ,$

where and 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 ''n'' is now countably infinite and the sequence of positive numbers accumulate only at 0, ''T'' is then a compact operator, and one has the canonical form for compact operators.

If the series Σ_''i'' ''α_i'' is convergent, ''T'' is a trace class operator.

Algebraic property

The family of finite-rank operators ''F''(''H'') on a Hilbert space ''H'' form a two-sided *-ideal in ''L''(''H''), the algebra of bounded operators on ''H''. In fact it is the minimal element among such ideals, that is, any two-sided *-ideal ''I'' in ''L''(''H'') must contain the finite-rank operators. This is not hard to prove. Take a non-zero operator ''T'' ∈ ''I'', then ''Tf'' = ''g'' for some ''f, g'' ≠ 0. It suffices to have that for any ''h, k'' ∈ ''H'', the rank-1 operator ''S''_{''h, k''} that maps ''h'' to ''k'' lies in ''I''. Define ''S''_{''h, f''} to be the rank-1 operator that maps ''h'' to ''f'', and ''S''_{''g, k''} analogously. Then

:$S_ = S_ T S_, \,$

which means ''S''_{''h, k''} is in ''I'' and this verifies the claim.

Some examples of two-sided *-ideals in ''L''(''H'') are the trace-class, Hilbert–Schmidt operators, and compact operators. ''F''(''H'') is dense in all three of these ideals, in their respective norms.

Since any two-sided ideal in ''L''(''H'') must contain ''F''(''H''), the algebra ''L''(''H'') is simple if and only if it is finite dimensional.

Finite-rank operators on a Banach space

A finite-rank operator $T:U\to V$ between Banach spaces is a bounded operator such that its range is finite dimensional. Just as in the Hilbert space case, it can be written in the form

:$T h = \sum _ ^n \langle u_i, h\rangle v_i \quad \mbox \quad h \in U ,$

where now $v_i\in V$, and $u_i\in U'$ are bounded linear functionals on the space $U$.

A bounded linear functional is a particular case of a finite-rank operator, namely of rank one.

References

{{DEFAULTSORT:Finite Rank Operator
Operator theory