Finite-rank operator

In functional analysis, a branch of mathematics, a finite-rank operator is a bounded linear operator between Banach spaces whose 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, has rank if and only if is of the form Exactly the same argument shows that an operator on a Hilbert space is of rank if and only if where the conditions on are the same as in the finite dimensional case. Therefore, by induction, an operator of finite rank takes the form 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 is now countably infinite and the sequence of positive numbers accumulate only at , is then a compact operator, and one has the canonical form for compact operators. If the series is convergent, is a trace class operator. The family of finite-rank operators on a Hilbert space form a two-sided *-ideal in , the algebra of bounded operators on . In fact it is the minimal element among such ideals, that is, any two-sided *-ideal in must contain the finite-rank operators. This is not hard to prove. Take a non-zero operator , then for some . It suffices to have that for any , the rank-1 operator that maps to lies in . Define to be the rank-1 operator that maps to , and analogously. Then which means is in and this verifies the claim. Some examples of two-sided *-ideals in are the trace-class, Hilbert–Schmidt operators, and compact operators. is dense in all three of these ideals, in their respective norms. Since any two-sided ideal in must contain , the algebra is simple if and only if it is finite dimensional. A finite-rank operator between Banach spaces is a bounded operator such that its range is finite dimensional.