Let T : B1 → B2 be an operator between Banach spaces. Then the transpose (or dual) of T is an operator satisfying:
for all x in B1 and y in B2*. Here, we used the notation: .
The necessary and sufficient condition for the transpose of T to exist is that T is densely defined (for essentially the same reason as to adjoints, as discussed above.)
For any Hilbert space H, there is the anti-linear isomorphism:
given by where . Through this isomorphism, the transpose T' relates to the adjoint T∗ in the following way:
where . (For the finite-dimensional case, this corresponds to the fact that the adjoint of a matrix is its conjugate transpose.) Note that this gives the definition of adjoint in terms of a transpose.
Read more about this topic: Closed Operator
Other articles related to "transpose":
... In the mathematical and algorithmic study of graph theory, the converse, transpose or reverse of a directed graph G is another directed graph on the same set of vertices with all of the ... That is, if G contains an edge (u,v) then the converse/transpose/reverse of G contains an edge (v,u) and vice versa ...
... Language/Library Create Determinant Transpose Element Column Row Eigenvalues Fortran m=RESHAPE(, SHAPE(m)) TRANSPOSE(m) m(i,j) m(,j) m(i,) Ch m ...
... such that PTAP = B where "T" denotes the matrix transpose ... Note that Halmos defines congruence in terms of conjugate transpose (with respect to a complex inner product space) rather than transpose, but this ...
... Ideally, one might hope to transpose a matrix with minimal additional storage ... This leads to the problem of transposing an n × m matrix in-place, with O(1) additional storage or at most storage much less than mn ...
Famous quotes containing the word transpose:
“We have to transpose ourselves into this impressionability of mind, into this sensitivity to tears and spiritual repentance, into this susceptibility, before we can judge how colorful and intensive life was then.”
—Johan Huizinga (18721945)