# Understanding literature - defining a matrix operation in a basis independent manner

I have been reading some notes: Vector Symmetries by Lyubashenko and I'm looking to understand some of the works to apply to my own research.

In it, he writes a matrix operation in a basis independent way so that it may be used in the context of operators. Essentially, he defines the "sharp" operation on a matrix $$M: \mathcal{V} \otimes \mathcal{V} \to \mathcal{V} \otimes \mathcal{V}$$ (where $$\mathcal{V}$$ is a finite dimensional vector space) as $$(M^{\#})^{\alpha \beta}_{\delta \gamma} = M^{\beta \gamma}_{\alpha \delta}$$

He then goes on to describe the same operation on an operator $$T: \mathcal{H} \otimes \mathcal{H} \to \mathcal{H} \otimes \mathcal{H}$$ ($$\mathcal{H}$$ a Hilbert space). He says "Let $$T^{\#} : \mathcal{H}\otimes \mathcal{H}^* \to \mathcal{H}^*\otimes \mathcal{H}$$ ($$\mathcal{H}^*$$ the duel space of $$\mathcal{H}$$) and let an element $$h_1^* \otimes h_2^* \otimes h_1 \otimes h_2 \in \mathcal{H}^* \otimes \mathcal{H}^* \otimes \mathcal{H}\otimes \mathcal{H}$$ represent $$T$$. After a cyclic rearrangement of factors, it yields $$h_2 \otimes h_1^* \otimes h_2^* \otimes h_1$$ that represents the operator $$T^{\#}$$ via $$T^{\#}(\eta \otimes \xi^*) = \langle h_2, \xi^*\rangle \langle \eta, h_1^*\rangle h_1^*\otimes h_2.$$ "

My questions are:

• What does it mean for an element to "represent an operator"?
• Does this "cyclic rearrangement of factors" corresponds to the 'rotation' of indices in the matrix case?
• How does this formula definition get applied in calculation? For example, we have the operator $$F$$ defined as $$F(\eta \otimes \xi) = \xi \otimes \eta$$. This operator should be invariant under the sharp operation, i.e $$F^{\#} = F$$, but how do we show this using the expression above?
• Furthermore, how is this definition consistent with the matrix case?

I apologise for the list of questions, and I'm very grateful for help.

• I know Volodymyr Lyubashenko. You can reach him via this data. – Alex Ravsky Apr 13 at 16:42