14.4.1 Tensor Linear Operator Product Factor Selection

Tensor Linear Operator Product Factor Selection selects factors to decompose operators, structuring complex linear transformations in tensor algebra.

Tensor Linear Operator Product Factor Selection is the process of choosing which operator acts on which tensor factor when forming an operator product f tensor g, together with the consequences that follow from selecting one factor to act nontrivially while the other is held fixed by the identity.

The Basic Selection

Choosing an Operator per Factor

Given operators f on V and g on W, factor selection is the act of assigning f to the first tensor factor and g to the second, producing

f \otimes g : V \otimes W \to V \otimes W .

Reversing the selection, so that g is assigned to the first factor and f to the second, produces instead g tensor f acting on W tensor V, a genuinely different operator on a differently ordered tensor product, unless a further identification between V tensor W and W tensor X is introduced.

Selecting the Identity for One Factor

A particular and frequently used factor selection assigns the identity operator to one of the two factors, giving

f \otimes {id}_{W} or {id}_{V} \otimes g,

selecting f to act alone on the first factor while leaving the second factor unaffected, or selecting g to act alone on the second factor while leaving the first factor unaffected.

Consequences of the Identity Selection

Restriction of Action to a Single Factor

When the identity is selected for the second factor, the elementary action reduces to

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

leaving w completely unchanged and transforming only the V component, so this factor selection produces an operator on V tensor W that behaves, on each fixed copy of V tensor w, exactly as f behaves on V alone.

Recombining Two Single-Factor Selections

Because the two factors act independently, selecting f for the first factor and identity for the second, followed by selecting identity for the first factor and g for the second, recombines through composition into the full operator product,

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

confirming that any factor selection assigning both f and g simultaneously can be built from two simpler, sequential single-factor selections in either order.

Selection in the Operator Algebra

Selection Determines Membership in Commuting Subalgebras

Selecting f for the first factor while fixing the identity on the second embeds f into a subalgebra of the endomorphism algebra of V tensor W, and selecting g for the second factor while fixing the identity on the first embeds g into a different subalgebra; the factor selection made for f and the factor selection made for g always produce commuting operators, since the two subalgebras commute elementwise regardless of which particular f and g are selected.

Selection and the Kronecker Product

Once bases are fixed, selecting f for the first factor and the identity for the second corresponds, in the induced basis, to the Kronecker product of the matrix of f with the identity matrix of size equal to the dimension of W, while selecting the identity for the first factor and g for the second corresponds to the Kronecker product of the identity matrix of size equal to the dimension of V with the matrix of g, giving explicit matrix forms for each single-factor selection.

Selection and Invariant Subspaces

Selection Preserving a Product Subspace

If U is a subspace of V invariant under f, then selecting f for the first factor and any operator g for the second factor produces an operator that maps U tensor W into U tensor W, since the elementary action sends elements of U tensor W to elements of U tensor W directly by the invariance of U under f, regardless of the selection made for the second factor.

Selection Restricted to a Fixed Complement

When V decomposes as a direct sum of an invariant subspace U and a complementary subspace, factor selection of f restricted to U, tensored with any selection for the second factor, describes the action of f tensor g on the corresponding piece U tensor W of the domain space, separately from its action on the complementary piece, allowing the operator to be analyzed one invariant piece at a time.