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### Infinite-dimensional analogue of orthogonal matrices

In functional analysis one learns that self-adjoint operators are the infinite dimensional generalisation of symmetric matrices and the dual operator is the generalisation of the transposed matrix ...
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In the book I read recently(Eberhand Zeidler, Applied Functional Analysis: Applications to Mathematical Physics (Applied Mathematical Sciences 108)), when consider the extension of symmetric operator ...
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On page 9 in http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.639.5952&rep=rep1&type=pdf it is being shown why the negative divergence is the adjoint of the gradient. $V: \mathbb R^n \... 1answer 28 views ### How can the adjoint be defined if$f$is not one to one Let$f \in \mathcal{L}(V, W)$. Moreover let's suppose$(e_1, ..., e_n)$is a basis of$V$and$f(e_i) = v_i$where the$v_i$aren't distinct (so there is at least$i \ne j$such that$v_i = v_j$) so ... 1answer 38 views ### How to use operators on tensor products This problem is from physics, but I have trouble understanding the math. I have some problem understanding how to use tensors. Let's say in Quantum Optics if I have the state in mode$b$(where I can ... 1answer 39 views ### Computing the Adjoint using the Definition$\langle Tv,w\rangle = \langle v, T^*w\rangle$Let$T$be the linear operator on$\mathbb{C}^2$defined by $$T(a,b)=(2ia+3b,a-b).$$ I am trying to compute the adjoint. The answer is $$T^*(c,d)=(-2ic+d,3c-d)\tag{1},$$ which can be seen by (i) ... 1answer 23 views ### Defining Hermitian Adjoints Non-degenerate Hermitian Forms that are NOT positive definite. I was looking around different textbooks and websites for the definition of a Hermitian adjoint. All the resources that I have checked including the one I am studying at the moment (Jeevanjee's Intro. ... 1answer 36 views ###$\underset{_{a\rightarrow 0}}{\lim }L_{a}^{\ast }\left( L_{a}L_{a}^{\ast }\right) ^{-1}L_{a}=?$Let$\varepsilon >0$and let$L$be a bounded operator acting on a Hibert space such that$L_{\lambda }=L-\lambda I$is surjective of every$\lambda $such that$\varepsilon >\left\vert \lambda \...
I want to prove that there exists always solutions to this equation $$g(x) = \int\limits_a^b {f(s,x)ds}$$ where $g \in {L^2}(0,1)$ is given and $f \in {L^2}((a,b) \times (0,1))$ is the uknown, \$0<...