Tagged Questions

31 views

I have a self-adjoint operator $d$ which acts on vector fields defined on $\mathbb{R}^n$. I am interested on its eigenvalues. That is, I study the equation $d(X)-\lambda X=0$. I have found that if ...
38 views

Periodic Laplace operator non closed in $C^2(0,L)$

How can I show that the Laplacian operator is not closed in the domain $D=\{f \in C^2(0,L) \mid \mbox{ f is vanishing in a neighborhood of 0 and L } \}$ for a fixed $L$? And how can I show that it is ...
65 views

If $\|Tv\|=\|T^*v\|$ for all $v\in V$, then $T$ is a normal operator

I have solved a question but I am not sure the last step of the question. If someone can verify it that would be great. Let $V$ be a finite dimensional vector space with complex inner product. Let ...
26 views

A : L₁→L₁ 1) A x=( x₁, x₂,.....xn , 0,0,....) 2) A x= (λ₁ x₁ ,λ₂ x₂,.....) |λ n|≤1 and λ n ∈ R I need to find adjoint of operators A in given space. ...
31 views

$T\colon L^2[0,1]→L^2[0,1]$ is given by $$Tx(t)=∫_0^1 tx(s)\,ds$$ How can we find adjoint operator of $T$ in this space? $\langle Tx,y\rangle= \langle x,T^*y\rangle$ should be okay.But what ...
77 views

Isometry <=> Adjoint left inverse [duplicate]

Is it true that: $$T\text{ isometric}\iff T^*\text{ left inverse}$$ Obviously: $$\text{"}\Rightarrow\text{": }\langle x,\tilde{x}\rangle=\langle Tx,T\tilde{x}\rangle=\langle x,T^*T\tilde{x}\rangle$$ ...
70 views

Is it true that: $$T\text{ isometric}\iff T^*\text{ left inverse}$$ Obviously: $$\text{"}\Rightarrow\text{": }\langle x,\tilde{x}\rangle=\langle Tx,T\tilde{x}\rangle=\langle x,T^*T\tilde{x}\rangle$$ ...
80 views

If $T\in B(H)$ for some Hilbert space $H$, is a normal operator and $T^2=I$, then $T=T^*$. It seemed simple when I first saw the claim, but I'm having trouble showing it. I know that it implies ...
46 views

Adjoint of unbounded Operators: Product and Sum

When precisely does equality hold for sum and product: $$S^*+T^*\subseteq (S+T)^*$$ $$S^*T^*\subseteq (TS)^*$$ So far I checked that for the sum the seemingly weaker condition implies the stronger ...
86 views

Let $(X,\langle\cdot,\cdot\rangle)$ be a Hilbert Space over $K$ with orthonormal basis $(x_n)$, and let $(\lambda_n)\in K$ be a bounded sequence. The mapping $T:X\to X$ is defined by ...
87 views

Adjoint of multiplication by $z$ in the Bergman space

I am learning Hilbert space theory from Halmos' "Introduction to Hilbert space and the theory of spectral multiplicity". While talking about understanding adjoints (p. 39), he calls special ...
71 views

Clarifying the definition of essential self-adjointness

If a Hilbert space operator $T$ has a unique self-adjoint extension, must the extension be the closure of $T$?
89 views

Suppose we have the following differential equation $$\psi''(y)- k^2 \psi(y) - \frac{U''(y)\phi(y)}{V(y)-c}=0$$ where $\psi$ and $\phi$ are complex valued functions, say on the interval $[0,1]$. ...
73 views

If an Operator $L$ is defined as $Lu=u''$ and $a_1u(0)+b_1u'(0)+c_1u(1)+d_1u'(1)=0$ along with $a_2u(0)+b_2u'(0)+c_2u(1)+d_2u'(1)=0$, then for what values of $a_1,b_1,c_1,etc$ is the operator ...
56 views

Help with a 'simple' sum of linear operators and their adjoints acting on an orthonormal basis

Given an orthonormal basis $\{u_1,\cdots, u_n\}$ of a vector space $V$ I am asked to show that $$\sum_{k=1}^n \|T^*u_k\|^2= \sum_{k=1}^n \|Tu_k\|^2$$ for all $T\in \mathcal{L}(V)$ where $T^*$ ...
64 views

$T:\ell_2\to \ell_2, T(x_1,x_2,\dotsc)=(x_1,x_2/2,x_3/3,\dotsc)$

$T:\ell_2\to \ell_2, T(x_1,x_2,\dotsc)=(x_1,x_2/2,x_3/3,\dotsc)$ I need to know whether it is self adjoint and unitary operator given that $x_i\in\mathbb C$ I am not able to do it please tell me how ...
138 views

Is it a unitary, self adjoint and normal operator?

Let $A\colon H\to H$ be a bounded linear operator on a complex Hilbert space such that $\|Ax\|=\|A^*x\|\forall x$, given that there is a nonzero $x$ for which $A^*x=(2+3i)x$. Then I need to know ...
36 views

self adjoint linear operator and integration

is this formula correct ?? $$\int_{-\infty}^{\infty} Lf(x)\delta (x-1)= \int_{-\infty}^{\infty} f(x)L^{\dagger}\delta(x-1)$$ here $L$ is a linear operator and $L^{\dagger}$ is its formal ...
59 views

Can I always extend a selfadjoint Operator in $L^2$?

Assume that we have a self-adjoint operator $T\colon D \to D$ where $D \subset L^2$ is some finite dimensional subspace. Can I conclude that than a self-adjoint operator $S \colon L^2 \to L^2$ exists ...
171 views

What is the adjoint of $x + \frac{d}{dx}?$

I have solved other problems like this using integration by parts. In this case, I can't figure out what to make each part for the integration. The question is true/false. Ultimately to show this you ...
59 views

Block Matrices of Operators

I'm trying to prove the following: Consider the vector space of matrices of size $n\times n$ whose entries in $\mathcal B(H)$. Denote this vector space by $M_{n,n}(\mathcal{B(H)})$. We can define ...
96 views

Showing an operator is self adjont

I am trying to show that the operator: $$Tf(s)=5s^2\int_0^1t^2f(t)dt+2\int_0^1f(t)dt$$ is self adjoint where $H=L(0,1)$ with real scalars and $t\in \mathcal{L}(H)$. So I can re-write this operator ...
101 views

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 ...
71 views

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?
228 views

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 ...
127 views

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 ...
79 views

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 $$... 2answers 234 views Find adjoint operator of an operator T I would like to find the adjoint operator of$$ T\colon L^2([0,1])\to H^1([0,1]),\quad x\mapsto\int\limits_0^t x(s)\, ds. $$Here H^1([0,1]) is the Sobolev space W^{1,2}([0,1]). I tried to find ... 1answer 160 views The convergence of the adjoint operator If a sequence of operator A_n converges in norm to A, i.e. \lim \lVert A_n-A\rVert=0)where A_n and A\in B(H) (H is the Hilbert space). Is it true that A_n^* converges in norm to A^*? 1answer 91 views Is \text{rk}L=\text{rk}L^*L  true for finite rank operators? Let L be a compact linear operator in an infinitedimensional space that has finite rank. Do the equations$$\text{rk}L=\text{rk}L^*L\ \text{and} \ \text{rk}L^*L=\text{rk}R, where $R$ is the ...
346 views

In abstract Hodge theory there is the following lemma: Let $H$ be a Hilbert space and $A \in \mathcal{C}(H)$ a densely defined, closed operator (so possibly unbounded) and $A^*$ its adjoint operator. ...
149 views

Norm of conjugate Hardy operator

For the classical Hardy operator $T\colon \ell^p\to \ell^p \quad (Tx)_n=\frac{1}{n}\sum_{k=1}^n x_k$ or the integral type $S\colon L^p\rightarrow L^p \quad (Sf)(x)=\frac{1}{x}\int_0^x f(t) dt \ \$ ...
Let $T: H \to H$ be a compact operator with $H$ a Hilbert space. Let then $\lambda \neq 0$ be an eigenvalue of $T$ with eigenfunction $v$. Is then $\lambda$ an eigenvalue for the adjoint $T^*$ ...