Miscellaneous questions on operators

[spaghetti3451]

Hey guys, I am wondering if the following relationships hold for all operators A, regardless of whether they are linear or non-linear.

A^-1A = AA^-1 = I

[A,B] = AB - BA

A|a_n> = λ_n|a_n>, where n ranges from 1 to N, and N is the dimension of the vector space which has an orthogonal basis |a_n>.

Just one other question. Which is the more general definition of the adjoint (hermitian conjugate) A† of an operator A: (v, Au) = (A†v, u) or A† = (A*)^T?

I think it's the first one. The second one is a special case of the first which is valid if the vectors v and u are matrices. Your thoughts?

Let's see if you can make a dumbass like me learn some maths!

 

[algebrat]

The first is def'n of inverse, which doesn't always exist.

The second is definition of commutator over the ring of operators.

The third is definition of eigenvalue and eigenfunction, and an assumption that there is a finite eigenbasis of the space. Note it is possible for some of the eigenvalues to be identical.

The fourth, your first expression is def'n of adjoint (the other of a mirrored pair, mirrored to other side of the inner product). But to say it's the transpose, you have to clarify the def'n of transpose. If A is a matrix, you're probably fine. If it's not, and say for example it's the derivative operator, then what do you mean by transpose. Also, if you're space is not flat, in other words, if there is some nontrivial metric, then the adjoint may get more complicated. See for example

wikipedia.org/wiki/Transpose#Transpose_of_linear_maps

and the issue of a metric tensor.

A course in Functional Analysis would treat most of this quantum mechanical material with some rigor.