14.11.3 Tensor Operator Product Codomain Tensor Space

The tensor operator product codomain tensor space defines the target space where tensor operators act, structuring tensor products and transformations in algebra.

Tensor Operator Product Codomain Tensor Space is the tensor product space that receives the output of a combined operator built from individual factor operators, formed as the tensor product of the codomains of each factor operator in the same order as the corresponding factors appear in the original tensor product.

Formation of the Codomain

Codomain of Each Factor Operator

Each factor operator entering the construction has its own codomain, which may coincide with its domain when the operator maps a factor space to itself, or may be a distinct space when the factor operator changes spaces.

T_{1} : V_{1} \to W_{1}

T_{2} : V_{2} \to W_{2}

The Combined Codomain

The combined operator built from these factor operators maps the tensor product of the domains to the tensor product of the codomains, so its codomain is fixed as soon as the codomains of the individual factor operators are fixed.

T_{1} \otimes T_{2} : V_{1} \otimes V_{2} \to W_{1} \otimes W_{2}

Structural Diagram of the Mapping

Domain and Codomain Tensor Spaces Side by Side

The diagram below shows the domain tensor space on the left mapping through the combined operator to the codomain tensor space on the right, with each factor space paired to its own image space.

Dimension of the Codomain Tensor Space

Multiplicative Dimension Relation

The dimension of the codomain tensor space equals the product of the dimensions of the individual codomain spaces of the factor operators, mirroring the same multiplicative relation that governs the domain tensor space.

\dim (W_{1} \otimes W_{2}) = \dim (W_{1}) \times \dim (W_{2})

Independence From the Domain Dimensions

The dimension of the codomain tensor space depends only on the codomains of the individual factor operators and not on the dimensions of their domains, so factor operators with very different domain sizes can still share the same codomain tensor space if their individual codomains match.

Basis Induced on the Codomain

Basis Vectors of the Codomain Tensor Space

Once a basis is fixed for each individual codomain space, the codomain tensor space inherits a basis consisting of all simple tensors formed by pairing one basis vector from each individual codomain, ordered according to the same convention used for the domain tensor space.

Coordinates of the Combined Operator's Output

The coordinates of the output of the combined operator, relative to this induced basis, are obtained directly from the matrix product representation of the combined operator applied to the coordinates of the input vector in the domain tensor space.

Special Cases of the Codomain

Operators That Preserve the Factor Spaces

When every factor operator maps its factor space to itself, the codomain tensor space coincides exactly with the domain tensor space, and the combined operator is an operator on a single tensor product space rather than a map between two different tensor product spaces.

Operators That Change Factor Spaces

When at least one factor operator maps its domain to a genuinely different codomain space, the codomain tensor space differs from the domain tensor space, and the combined operator is properly viewed as a map between two distinct tensor product spaces.

Extension to Several Factors

Codomain for a Product of Many Factor Operators

When the combined operator is built from three or more factor operators, its codomain tensor space is the tensor product of all the individual codomain spaces, taken in the same order as the corresponding factor operators appear in the construction.

Consistency With Grouping of Factors

Regardless of how the factor spaces are grouped when forming the repeated tensor product, the resulting codomain tensor space is the same, since the tensor product of the individual codomains does not depend on the order in which the factors are associated together.