1,061 questions
Filter by
Sorted by
Tagged with
22k views

• 3,027
786 views

### Hermitian Operators and the Spectral Theorem

I understand that in a finite-dimensional vector space $V$, a diagonalizable linear operator $T: V \to V$ decomposes $V$ into a direct sum of its invariant eigenspaces, on each of which it restricts ...
802 views

### When is the restriction of a normal operator not normal?

I was proving the spectral theorem for normal operators on finite-dimensional complex vector spaces today during a test, when I arrived at the point in which If $T\in\operatorname{End}(V)$ is ...
• 903
5k views

### Adjoint of an integral operator

I'm reading through a text about integral operators and I've come across the following theorem: Let $k:\mathbb{R}^2\rightarrow\mathbb{C}$ be a kernel, $T:L^2(\mathbb{R})\rightarrow L^2(\mathbb{R})$ ...
• 27.9k
938 views

### Adjoint of a nonlinear operator?

For any linear operator $A$, the adjoint $A^*$ is defined as a linear operator that satisfies $$\langle v, Au\rangle = \langle A^*v, u\rangle$$ Moreover, one has that $(A^{*})^{*} = A$ (proof here). ...
• 1,584
4k views

### Help Showing that the Adjoint Operator $T^*$ is Surjective if and only if $T$ is Injective

Let $T\in L(V,W)$,where $L(V,W)$ denotes a linear map from a vector space $V$ to vector space $W$. I want to prove that $T$ is injective iff $T^*$ is surjective, where $T^*$ is the adjoint of $T$. I ...
1k 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 ...
• 323
710 views

### Why are self-adjoint operators important?

I am learning about self-adjoint and normal operators. So far, they have come up in the Spectral theorem, which says self-adjoint operators have an eigenvalue basis and a corresponding diagonal ...
• 1,716
1k views

### Schur decomposition of a matrix with distinct eigenvalues is almost unique

Let $M\in \mathbb C^{n,n}$ have $n$ distinct eigenvalues, and let $U_1, U_2$ be two Schur-forms of $M$. Show that if $U_1, U_2$ have equal diagonals, there is a hermitian diagonal matrix $Q$ such that ...
• 7,480
794 views

### Adjoint of multiplication by $z$ in a Hilbert Space (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 ...
• 3,257
703 views

### Exponential Law for based spaces

I realize most people work in "convenient categories" where this is not an issue. In most topology books there is a proof of the fact that there is a natural homeomorphism of function spaces (with ...
• 1,486
1k views

I hope this is enough of a question (and less of a start of a discussion) to be allowed here. I am trying to get my head around the ideas of inverse and adjoint operators. To keep it simple, let's ...
• 502
519 views

### Self adjoint operators on a Hilbert space

Let $H$ be a Hilbert space and let $T\in \mathcal{B}(H)$ such that $T$ is self-adjoint. I want to show that if $T$ is non-zero, then $T^n\neq 0$ for all $n\in \mathbb{N}$. Suppose $n$ be the least ...
• 4,744
826 views

### General Cauchy-Schwarz for adjoint positive operators

I'm trying to prove the next inequality, like Cauchy-Schwarz standard inequality: $$|\langle Tx,y\rangle |\leq\langle Tx,x\rangle ^{1/2}\langle Ty,y\rangle ^{1/2}\space\forall x,y\in\mathcal{H},$$ ...
• 1,267
1k views

### Connection between categorical notion of adjunction and dual space/adjoint in vector spaces

I'm an economist, not a mathematician. I've been trying to make sense of some concepts in functional analysis: dual, bidual, adjoint, natural mapping. The definitions of these notions come out of ...
2k views

### Invertibility in a finite-dimensional inner product space

Let $T$ be an invertible linear operator on a finite-dimensional inner product space. I just want a hint as to how I should prove that $T^{*}$ is also invertible and $( T^{-1} )^{*} = ( T^{*} )^{-1}$. ...
• 321
343 views

### Spectral gap and Poincaré inequality

Consider the PDE $$\partial_t u = L u$$ where $L = \Delta + \nabla V \cdot \nabla$ is a self-adjoint operator. I read that if $L$ has a spectral gap $\lambda > 0$ then "[convergence of the ...
• 1,684
903 views

### Derivative of inner product via adjoint operator vs. complex derivatives

Dear math enthusiasts, I need to take the derivative of an inner product involving an operator on one side and I'd like to do this via the adjoint operator. However, it seems I'm doing something ...
• 2,480
656 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 ...
184 views

### Why is the Compactness of an Operator so important? What is the use of compact operators in Mathematics?

Compact Operators have been the major topic in our Operator Theory course for the past few weeks. All the theorems which tell us whether a operator is compact or not are clear to me, but I still don't ...
• 537
366 views

• 633
188 views

Let $(M,g)$ be a Riemannian manifold and let $E \to M$ be a real vector bundle over $M$. Let $d_A = d+A$ be a covariant derivative on $E$. It is an $\mathbb R$-linear map $d_A \colon \Gamma(E) \to \... • 8,326 5 votes 1 answer 138 views ### Show properties of the linear operator$L((a_n)_n)=\left(\frac{1}{n}*a_n\right)_n$Let$L\colon \ell^p \rightarrow \ell^p$such that$L((a_n)_n)= \left(\frac{1}{n}*a_n\right)_n$. 1) Determine$L'$(adjoint operator),$\ker(L)$,$\ker(L')$,$\operatorname{rg}(L)$,$\operatorname{rg}(...
• 411
1k 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. ...
• 1,134
679 views

### Prove or disprove: $\operatorname{Adj} (A)$ is diagonlizable $\implies A$ is diagonalizable

For $2X2$: $$A:\\ \begin{bmatrix} a & b \\ c & d \end{bmatrix}$$ $$\operatorname{Adj}(A):\\ \begin{bmatrix} d & -c \\ -b & a \end{bmatrix}$$ So the statement is true. The ...
• 1,301
6k views

### Derivation of Adjoint for SO(3)

I am in the process of learning about Lie Algebras and Lie groups, specifically for $SO(3)$ and $SE(3)$. I've been reading a tutorial here: http://www.ethaneade.org/lie.pdf, but I'm getting stuck at ...
• 41