1,061 questions
Filter by
Sorted by
Tagged with
1 vote
8 views

Eigenvalues of an operator and its adjoint

I would like to verify the proof given in an answer #2 form here. The claim to prove was that $\lambda$ is an eigenvalue of $T$ if and only if $\overline{\lambda}$ is eigenvalue of the adjoint ...
81 views

Change of basis and diagonalization

For my notation here I'm going to use $'$ to indicate a new basis. Let $U$ be a unitary operator that takes a basis vector $|i\rangle$ and transforms it to a new basis vector $|i'\rangle$. So let's ...
1 vote
35 views

Find self-adjoint of $P=|a \rangle \langle b |$

Let $P$ be an operator s.a. $P=|a \rangle \langle b |$ and $P|f \rangle = \langle b | f \rangle | a \rangle$. Find the self-adjoint and the $P^2$ operator. My attempt: We know to find the self ...
30 views

Question about operators from $\mathbb{R}^\mathbb{N}$ to/from a real separable Hilbert space.

Let $X \equiv \mathbb{R}^\mathbb{N}$ denote the space of real sequences with the product topology. Suppose that $T \colon \mathcal{H} \to X$ is a linear operator, where $\mathcal{H}$ is a real, ...
32 views

Let $E$ be a Euclidean Space and $u\in\mathcal L(E)$. We let $u^{\star}\in\mathcal L(E)$ such that $\forall x,y\in E, \langle u(x)|y\rangle = \langle x|u^{\star}(y)\rangle$. We want to show that if $u\... 0 votes 0 answers 28 views Derivative self-adjoint operator 1)I have the symbol$\circ$? In$-(\mathcal{L} \psi)\circ \psi^{-1}$How to read it in math in this case ? 2)The derivative of$q(y)=-(\mathcal{L} \psi(x)) $is$q^\prime(y)= \frac{-{[\mathcal{L}...
196 views

Consider an infinite dimensional complex Hilbert space $H$. I think that for a bounded operator-valued function $A: x\mapsto A(x) \in \mathcal B(H)$, where $x\in \mathbb R$, we can define the ...
1 vote
1k views

40 views

109 views

Definition of the formal $L^2$-adjoint $T^*$ of a linear operator $T:C^\infty(T^*M\odot T^*M)\to C^\infty(M)$

Let $(M,g)$ be a Riemannian manifold, $C^\infty(T^*M\odot T^*M)$ the space of all smooth symmetric $2$-tensor fields on $M$, and $C^\infty(M)$ the space of all smooth functions on $M$. I'd like to ...
23 views

A question about weak solution of the adjoint operator in Evans' PDE

In Partial Differential Equations (Evans, 2nd edition) $\S$6.2.3, the author discusses about the Fredholm alternative w.r.t the second order elliptic PDE. My question may require that you are familiar ...
291 views

Matrix of self-adjoint operator such that every element of the diagonal is $0$.

Let $V$ be a finite dimensional $\mathbb R$-vector space and let $T:V\rightarrow V$ be an self-adjoint operator such that $\text{trace}(T)=0$. Show that there exists an orthonormal basis $B$ such that ...
254 views

Proving $\dim\ker(I-T)=\dim\ker(I-T^*)<\infty$ for finite-rank operator on a Hilbert space

In my functional analysis class, I have encountered the following problem Let $H$ be a Hilbert space and $T$ a finite-rank (its range is finite-dimensional) and bounded linear operator on $H$. We are ...
1 vote
16 views

87 views

Bounded below adjoint operator on the dual of an ordered Banach space

Suppose $X$ is a real Banach space with a $\textit{generating}$ closed cone $X_+$. That is $X=X_+ - X_+$. Let $B\subset X$ be the $\textit{open}$ unit ball, and denote $B_+:=B\cap X_+$. Show that the ...
59 views

Closed range of adjoint operator: Closed range theorem and a theorem in Conway's Functional Analysis book

Recently I came across the "Closed range theorem, which applies to densely defined closed operators. However, I'm also aware of a similar theorem (VI.1.10) in Conway's A Course in Functional ...
1 vote
78 views

Proof that the orthonormal projection $P$ onto $W$ in a Hilbert space satisfies $P^2=P$ and $P*=P$.

Let $H$ be a Hilbert space. Given a closed subspace $W\subseteq H$, the orthogonal projection onto $W$ is the unique bounded linear operator $P$ such that $\text{Im}(P)=W$ and $\ker(P)=W^{\perp}$. The ...
16 views

If $A \in \Bbb C^{n\times n}$, Show that $L(A, b) \neq \emptyset$ iff $b \in L(A^H, 0)^\perp$

If I put the problem into other words, then $Ax = b$ has a solution iff b is in the orthogonal complement of the kernel of $A^*$ i.e. the complex transpose of A. I know how to do these problems when ...
2k views

1 vote
21 views

Finding adjoint of a Matrix equation

For $\alpha_i \in \mathbb{C}$, write down the adjoint of \alpha_1\left|V_1\right\rangle=\alpha_2 \Theta \Pi\left|V_2\right\rangle\left\langle V_3 \mid V_4\right\rangle+\alpha_3^*\left|V_5\right\...
26 views

Adjoint of a Densely Defined Unbounded Operator is Unique

Let $\mathcal H$ be a Hilbert space and $A: \mathcal D_A \to \mathcal H$ be an unbounded linear operator. Suppose also that $\mathcal D_A$ is dense in $\mathcal H$. We define the adjoint of $A$, $A^*$ ...
1k views

A linear operator $T$ on an inner product space $V$ is said to have an adjoint operator $T^*$ on $V$ if $⟨T(u),v⟩=⟨u,T^*(v)⟩$ for every $u,v\in V$ I know how to proof "why this operator exist$(\text{... 2 votes 0 answers 27 views Bounded surjective linear map from$L^p$to$L^r$with$r < p$I am thinking about the following problem. Let$1 \leq r < p \leq \infty$, and consider the spaces$L^r([0,1])$and$L^p([0,1])$. A routine application of Holder shows that$L^p([0,1]) \subset L^r([...
Determine whether the linear operator $T(f) = f'$ (taking the derivative) has an adjoint or not. Consider the inner product $\left<f,g\right> = \int_0^1 f(t)g(t)\ dt$ defined on the vector ...
The question asks to find the adjoint of the operator T on P$_1$([0,1]), which is the space of polynomials with degree no greater than 1 over the field [0,1], which contains all numbers between 0 and ...