Finite rank operator

In functional analysis, a finite rank operator is a bounded linear operator between Banach spaces whose range is finite dimensional. For functional analysis as used in psychology see the Functional analysis (psychology article In Functional analysis (a branch of Mathematics) a bounded linear operator is a Linear transformation L between Normed vector spaces In Mathematics, Banach spaces (ˈbanax named after Polish Mathematician Stefan Banach) are one of the central objects of study in Functional analysis In Mathematics, the range of a function is the set of all "output" values produced by that function

1 Finite rank operators on a Hilbert space
- 1.1 A canonical form
- 1.2 Algebraic property
2 Finite rank operators on a Banach space

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, they can 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

$M = \alpha \cdot u v^*, \quad \mbox{where} \quad \|u \| = \|v\| = 1 \quad \mbox{and} \quad \alpha \geq 0 .$

Exactly the same argument shows that an operator T on a Hilbert space H is rank 1 if and only if

$T h = \alpha \langle h, v\rangle u \quad \mbox{for all} \quad h \in H ,$

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

$T h = \sum _{i = 1} ^n \alpha_i \langle h, v_i\rangle u_i \quad \mbox{for all} \quad h \in H ,$

where {u_i} and {v_i} are orthonormal bases. Notice this is essentially a restatement of singular value decomposition. In Linear algebra, the singular value decomposition ( SVD) is an important factorization of a rectangular real or complex matrix 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 {α_i} accumulate only at 0, T is then a compact operator, and one has the canonical form for compact operators. In Functional analysis, Compact operators on Hilbert spaces are a direct extension of matrices in the Hilbert spaces they are precisely the closure of Finite

If the series ∑_i α_i is convergent, T is a trace class operator. In Mathematics, a trace class operator is a Compact operator for which a trace may be defined such that the trace is finite and independent of the choice

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 surffices 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

$S_{h,k} = S_{g,k} T S_{h,f}, \,$

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. In Mathematics, a trace class operator is a Compact operator for which a trace may be defined such that the trace is finite and independent of the choice In Mathematics, a Hilbert–Schmidt operator is a Bounded operator A on a Hilbert space H with finite Hilbert–Schmidt norm In Functional analysis, a branch of Mathematics, a compact operator is a Linear operator L from a Banach space X to another 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. In Mathematics, specifically in Ring theory, an algebra is simple if it contains no non-trivial two-sided ideals and the set {

Finite rank operators on a Banach space

Finite rank operator $T:U\to V$ between Banach spaces is a bounded operator such that its range is finite dimensional. In Mathematics, Banach spaces (ˈbanax named after Polish Mathematician Stefan Banach) are one of the central objects of study in Functional analysis In Functional analysis (a branch of Mathematics) a bounded linear operator is a Linear transformation L between Normed vector spaces In Mathematics, the range of a function is the set of all "output" values produced by that function Just as in the Hilbert space case, it can be written in the form

$T h = \sum _{i = 1} ^n \alpha_i \langle h, v_i\rangle u_i \quad \mbox{for all} \quad h \in U ,$

where now $u_i\in V$ , and $v_i\in U'$ 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.

© 2009 citizendia.org; parts available under the terms of GNU Free Documentation License, from http://en.wikipedia.org
network: | |

Contents

Finite rank operators on a Hilbert space

A canonical form

Algebraic property

Finite rank operators on a Banach space