GoI for MELL: partial isometries
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Operators, partial isometries
We will denote by H the Hilbert space of sequences of complex numbers such that the series converges. If and are two vectors of H their scalar product is:
- .
Two vectors of H are othogonal if their scalar product is nul. We will say that two subspaces are disjoint when any two vectors taken in each subspace are orthorgonal. Note that this notion is different from the set theoretic one, in particular two disjoint subspaces always have exactly one vector in common: 0.
The norm of a vector is the square root of the scalar product with itself:
- .
Let us denote by the canonical hilbertian basis of H: where δ_{kn} is the Kroenecker symbol: δ_{kn} = 1 if k = n, 0 otherwise. Thus if is a sequence in H we have:
- .
An operator on H is a continuous linear map from H to H.^{[1]}The set of (bounded) operators is denoted by .
The range or codomain of the operator u is the set of images of vectors; the kernel of u is the set of vectors that are anihilated by u; the domain of u is the set of vectors orthogonal to the kernel, ie, the maximal subspace disjoint with the kernel:
- ;
- ;
- .
These three sets are closed subspaces of H.
The adjoint of an operator u is the operator u^{ * } defined by for any . Adjointness is well behaved w.r.t. composition of operators:
- (uv)^{ * } = v^{ * }u^{ * }.
A projector is an idempotent operator of norm 0 (the projector on the null subspace) or 1, that is an operator p such that p^{2} = p and or 1. A projector is auto-adjoint and its domain is equal to its codomain.
A partial isometry is an operator u satisfying uu^{ * }u = u; this condition entails that we also have u^{ * }uu^{ * } = u^{ * }. As a consequence uu^{ * } and uu^{ * } are both projectors, called respectively the initial and the final projector of u because their (co)domains are respectively the domain and the codomain of u:
- Dom(u^{ * }u) = Codom(u^{ * }u) = Dom(u);
- Dom(uu^{ * }) = Codom(uu^{ * }) = Codom(u).
The restriction of u to its domain is an isometry. Projectors are particular examples of partial isometries.
If u is a partial isometry then u^{ * } is also a partial isometry the domain of which is the codomain of u and the codomain of which is the domain of u.
If the domain of u is H that is if u^{ * }u = 1 we say that u has full domain, and similarly for codomain. If u and v are two partial isometries then we have:
- uv^{ * } = 0 iff u^{ * }uv^{ * }v = 0 iff the domains of u and v are disjoint;
- u^{ * }v = 0 iff uu^{ * }vv^{ * } = 0 iff the codomains of u and v are disjoint;
- uu^{ * } + vv^{ * } = 1 iff the codomains of u and v are disjoint and their their direct sum is H.
Partial permutations
We will now define our proof space which turns out to be the set of partial isometries acting as permutations on the canonical basis .
More precisely a partial permutation on is a one-to-one map defined on a subset of onto a subset of . is called the domain of and its codomain. Partial permutations may be composed: if ψ is another partial permutation on then is defined by:
- iff and ;
- if then ;
- the codomain of is the image of the domain: .
Partial permutations are well known to form a structure of inverse monoid that we detail now.
Given a a subset D of , the partial identity on D is the partial permutation defined by:
- ;
- for any .
Thus the codomain of is D.
The partial identity on D will be denoted by 1_{D}. Partial identities are idempotent for composition.
Among partial identities one finds the identity on the empty subset, that is the empty map, that we will denote by 0 and the identity on that we will denote by 1. This latter permutation is the neutral for composition.
If is a partial permutation there is an inverse partial permutation whose domain is and who satisfies:
The proof space
Given a partial permutation one defines a partial isometry by:
In other terms if is a sequence in then is the sequence defined by:
- if , 0 otherwise.
We will (not so abusively) write when is undefined so that the definition of reads:
- .
The domain of is the subspace spanned by the family and the codomain of is the subspace spanned by . In particular if is 1_{D} then is the projector on the subspace spanned by .
Definition
We call p-isometry a partial isometry of the form where is a partial permutation on . The proof space is the set of all p-isometries.
Proposition
Let and ψ be two partial permutations. We have:
- .
The adjoint of is:
- .
In particular the initial projector of is given by:
- .
and the final projector of is:
- .
If p is a projector in then there is a partial identity 1_{D} such that .
Projectors commute, in particular we have:
- .
Note that this entails all the other commutations of projectors: and .
In particular note that 0 is a p-isometry. The set is a submonoid of but it is not a subalgebra.^{[2]}In general given we don't necessarily have . However we have:
Proposition
Let . Then iff u and v have disjoint domains and disjoint codomains, that is:
- iff uu^{ * }vv^{ * } = u^{ * }uv^{ * }v = 0.
Proof. Suppose for contradiction that e_{n} is in the domains of u and v. There are integers p and q such that u(e_{n}) = e_{p} and v(e_{n}) = e_{q} thus (u + v)(e_{n}) = e_{p} + e_{q} which is not a basis vector; therefore u + v is not a p-permutation.
As a corollary note that if u + v = 0 then u = v = 0.
From operators to matrices: internalization/externalization
It will be convenient to view operators on H as acting on , and conversely. For this purpose we define an isomorphism by where and are partial isometries given by:
- p(e_{n}) = e_{2n},
- q(e_{n}) = e_{2n + 1}.
From the definition p and q have full domain, that is satisfy p^{ * }p = q^{ * }q = 1. On the other hand their codomains are disjoint, thus we have p^{ * }q = q^{ * }p = 0. As the sum of their codomains is the full space H we also have pp^{ * } + qq^{ * } = 1.
Note that we have choosen p and q in . However the choice is arbitrary: any two p-isometries with full domain and disjoint codomains would do the job.
Given an operator u on H we may externalize it obtaining an operator U on defined by the matrix:
where the u_{ij}'s are given by:
- u_{11} = p^{ * }up;
- u_{12} = p^{ * }uq;
- u_{21} = q^{ * }up;
- u_{22} = q^{ * }uq.
The u_{ij}'s are called the external components of u. The externalization is functorial in the sense that if v is another operator externalized as:
then the externalization of uv is the matrix product UV.
As pp^{ * } + qq^{ * } = 1 we have:
- u = (pp^{ * } + qq^{ * })u(pp^{ * } + qq^{ * }) = pu_{11}p^{ * } + pu_{12}q^{ * } + qu_{21}p^{ * } + qu_{22}q^{ * }
which entails that externalization is reversible, its converse being called internalization.
If we suppose that u is a p-isometry then so are the components u_{ij}'s. Thus the formula above entails that the four terms of the sum have pairwise disjoint domains and pairwise disjoint codomains from which we deduce:
Proposition
If u is a p-isometry and u_{ij} are its external components then:
- u_{1j} and u_{2j} have disjoint domains, that is for j = 1,2;
- u_{i1} and u_{i2} have disjoint codomains, that is for i = 1,2.
As an example of computation in let us check that the product of the final projectors of pu_{11}p^{ * } and pu_{12}q^{ * } is null:
where we used the fact that all projectors in commute, which is in particular the case of pp^{ * } and u^{ * }pp^{ * }u.
Notes and references
- ↑ Continuity is equivalent to the fact that operators are bounded, which means that one may define the norm of an operator u as the sup on the unit ball of the norms of its values:
- .
- ↑ is the normalizing groupoid of the maximal commutative subalgebra of consisiting of all operators diagonalizable in the canonical basis.