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### Invertible operator norm bound

Let $H$ be a Hilbert space and that $X$ are bounded. Suppose $X$ is self-adjoint. Show that $Y=X+iI$ is invertible and the inverse $Y^{-1}$ has the norm $\lVert Y^{-1} \rVert \le 1$. I can prove $Y$ ...
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### System of equations wrt self-adjoint operators

$X = \left( \begin{matrix} 2&s\\ 8&2 \end{matrix} \right)$ and $Y = \left( \begin{matrix} 2&-1\\ 2&2 \end{matrix} \right)$ are two operators wrt the same orthonormal basis $B$ in a 2D ...
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### Operator inequalities: $0 \leq A \leq B \Rightarrow Tr(A^p) \leq Tr(B^p)$?

It is trivial to show that $0 \leq A \leq B \Rightarrow Tr(A^2) \leq Tr(B^2)$, but does this generally hold for all $p >$ 2 as well?
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### Prove that every self-adjoint unitary linear operator can be expressed in the form $U\alpha = \beta - \gamma$

This problem is from Kunze Hoffman book. I think I go in the right direction to solve this but I miss some point to finish it. Can anyone help me? Suppose $U$ is a self-adjoint unitary linear ...
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This is about, a question I answered. Now there is an additional question that I cannot answer and do not want to spend any more time on. I feel like the question will not get any attention, as I ...
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I would like to find the adjoint operator in the Hilbertspace $L^2(0,\infty)$ of the operator $$(Ax)(t)=x(at), x\in L^2(0,\infty), a>0.$$ My calculation is the following; I use the ...
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### Multiplicationoperator self-adjoint on given domain

I have a question. Given the Hilbertspace $H=l^2(\Bbb{N})$ with multiplicationoperator $T_f$, with $f:\Bbb{N}\rightarrow\Bbb{C}$ and $T_f\psi=f\psi$. I want to prove the following statement: Suppose ...
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Show that the Singular Value Decomposition of the operator $$A\colon L^2([0,1])\to L^2([0,1]), x\mapsto\int\limits_0^t x(s)\, ds$$ is given by $$\sigma_j=\frac{1}{(j-1/2)\pi},~~~~~v_j(x)=\sqrt{2}... 2answers 527 views ### Find adjoint operator of an operator T I would like to find the adjoint operator of$$ T\colon L^2([0,1])\to H^1([0,1]),\quad x\mapsto\int\limits_0^t x(s)\, ds. $$Here H^1([0,1]) is the Sobolev space W^{1,2}([0,1]). I tried to find ... 1answer 3k views ### Why is every selfadjoint operator closed? I've read this theorem multiple times, but never seen a proof: Every selfadjoint operator is closed. But it's always been stated without a proof. Is it somehow obvious? I can't see it immediately ... 2answers 53 views ### Using “adjunction” to refer to the act of taking adjoints of operators I have an especially flabby terminology question. How acceptable is it, in your opinion, to use the word "adjunction" to refer to the process of taking adjoints of operators on a Hilbert space? ... 2answers 1k views ### Is there a general expression for the adjoint representation of U(N) or u(N)? At least for low values of N like 2 or 3 and such I would like to know if there are explicit matrices known giving the representation of u(N) or U(N) in the adjoint? (..a related query: ... 1answer 456 views ### The convergence of the adjoint operator If a sequence of operator A_n converges in norm to A, i.e. \lim \lVert A_n-A\rVert=0)where A_n and A\in B(H) (H is the Hilbert space). Is it true that A_n^* converges in norm to A^*? 1answer 207 views ### Verifying the Transpose of a Linear Operator I have a linear operator \Phi in \mathbb{R}^n and have created another operator that I believe to be its adjoint (transpose) \Phi^T. What is the most direct way to verify that my \Phi^T is ... 2answers 1k views ### Hilbert Adjoint Operator from Riesz Representation Theorem - T^{*}y=\frac{\left\langle y,Tx\right\rangle }{\left\langle z_{0},z_{0}\right\rangle}z_0 Kreyszig's Functional analysis seems to introduce the hilbert adjoint operator by means of an explicit representation. I haven't seen this anywhere else and I would like to confirm this explicit ... 1answer 209 views ### Invertibility of matrix with each element equal to cofactor I am doing an exercise book which has one problem that asks you to prove the nonsingularity of a matrix if each element of the matrix equals its cofactor (the determinant submatrix by deleting the ... 1answer 544 views ### self-adjoint operator and unitary orthogonal matrix Please offer a solution to the following problem. It was offered in class by my professor as an additional exercise to try on one's own. Let V be the inner product space, and assume that \alpha \... 1answer 562 views ### Find an adjoint; inner product Let V be the space of real polynomials of degree at most 1 with inner product defined by \langle p,q \rangle = \frac12\int_{-1}^1p(t)q(t)dt. Define \alpha\in End(V) by$$\alpha(p)=p(0)+p(1)t.$$... 2answers 958 views ### inner product and adjoint operator This is a problem I found in Schaum's Outlines: Linear Algebra, and I was wondering if someone knew how to solve it. I began using integration by parts, but that approach did not lead to any ... 1answer 6k views ### Cofactor matrix 4x4, evaluated by hand I am currently studying linear algebra. So far I have studied method for aquiring the inverse of a matrix A. Now I would like to evaluate the inverse of a 4\times 4 matrix using the following ... 1answer 119 views ### Is \text{rk}L=\text{rk}L^*L  true for finite rank operators? Let L be a compact linear operator in an infinitedimensional space that has finite rank. Do the equations$$\text{rk}L=\text{rk}L^*L\ \text{and} \ \text{rk}L^*L=\text{rk}R, where $R$ is the (...
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Recently I found this statement -- the gradient operator is the adjoint of the minus divergence operator -- in one of my lecture notes. Knowing only a little about functional analysis, I'm looking for ...