14.4.5 Tensor Linear Operator Product Extension Rule

The Tensor Linear Operator Product Extension Rule extends linear operators on tensors, defining how they act on multi-linear structures through tensor product spaces.

Tensor Linear Operator Product Extension Rule is the principle governing how a single operator defined on one factor of a tensor product is extended to an operator on the whole tensor product by pairing it with the identity operator on the remaining factor.

Statement of the Extension Rule

Extending an Operator on the First Factor

Given an operator f on V and a fixed space W, the extension rule produces the operator

f \otimes {id}_{W} : V \otimes W \to V \otimes W,

acting on elementary tensors by

(f \otimes {id}_{W}) (v \otimes w) = f (v) \otimes w,

so that f acts exactly as before on the V component while the W component passes through unchanged.

Extending an Operator on the Second Factor

Symmetrically, given an operator g on W and a fixed space V, the extension rule produces

{id}_{V} \otimes g (v \otimes w) = v \otimes g (w),

leaving the V component unchanged while g acts on the W component.

Why the Extension Rule Is Well Defined

Compatibility with the General Construction

The extension rule is a direct special case of the general tensor product of maps, obtained by choosing the second map in the factor map pair to be an identity operator, so the well-definedness of f tensor identity-on-W is guaranteed by the same universal property argument that guarantees well-definedness of the general tensor product of maps, requiring no separate justification.

Linearity of the Extended Operator

Because the extended operator arises from the universal property, it automatically respects addition and scalar multiplication on the whole of V tensor W, not merely on elementary tensors, matching the linearity preservation already established for tensor products of maps in general.

Consequences of the Extension Rule

Recovering the Full Operator Product

The extension rule provides the building blocks for the full operator product through the factorization

f \otimes g = (f \otimes {id}_{W}) \circ ({id}_{V} \otimes g),

showing that any operator product can be assembled from two applications of the extension rule, one for each factor, composed in either order.

Commutativity of the Two Extensions

The two extensions built from the extension rule always commute with one another,

(f \otimes {id}_{W}) \circ ({id}_{V} \otimes g) = ({id}_{V} \otimes g) \circ (f \otimes {id}_{W}),

regardless of the choice of f and g, since the composition identity for tensor products of maps reduces both sides to f tensor g directly.

Extension Rule and Spectral Data

Eigenvalues Under the Extension Rule

If lambda is an eigenvalue of f with eigenvector u, then lambda is also an eigenvalue of f tensor identity-on-W, with eigenspace containing u tensor w for every w in W, since

(f \otimes {id}_{W}) (u \otimes w) = (λ u) \otimes w = λ (u \otimes w),

so the eigenvalue lambda of f is inherited by the extension rule with multiplicity multiplied by the full dimension of W, since every choice of w in W contributes an independent eigenvector.

Invariant Subspaces Under the Extension Rule

If U is a subspace of V invariant under f, the extension rule sends the subspace U tensor W of V tensor W into itself, since the extension acts trivially on the W component and preserves U on the V component, giving a direct correspondence between invariant subspaces of f alone and invariant subspaces of its extension of the form U tensor W.