1,167 questions
Filter by
Sorted by
Tagged with
37 views

### Let $H$ be a Hilbert space and let $A \in \mathcal{B}(H)$ be such an operator that $A^3 = A^*$.

Let $H$ be a Hilbert space and let $A \in \mathcal{B}(H)$ be such an operator that $A^3 = A^*$. (a) Show that from $A^4 = 0$ it follows $A = 0$. (b) Find the eigenvalues of the operator $A$. (c) ...
• 101
34 views

• 2,612
29 views

### Let $H$ be a Hilbert space, $A, B \in \mathcal{B}(H)$, and let $C = A^*A + B^*B$.

Let $H$ be a Hilbert space, $A, B \in \mathcal{B}(H)$, and let $C = A^*A + B^*B$. Show: (a) $\ker C = \ker A \cap \ker B$. (b) The eigenvalues of the operator $C$ are non-negative real numbers. ...
• 365
30 views

• 365
33 views

I encountered a math problem when reading some optimization papers. In one of the paper, it minimize a function of Frobenius norm of complex matrices: $\min_{U} \frac{\beta}{2}\mathrm{\left\| H-UV_{}^{... -2 votes 0 answers 38 views ### Question about uniqueness of adjoint operator Let$S, S'\in L(V)$be such that $$\langle Tx,y\rangle=\langle x, Sy\rangle=\langle x,S'y\rangle$$ for all$x,y\in V$. Defining$C=S-S'\in L(V)$, we have $$\langle x, Cy\rangle=\langle x, S-S'y\rangle ... • 1,420 0 votes 0 answers 30 views ### Is this differential operator self-adjoint? If V is the space of functions in C^{\infty}(\mathbb{R}) such that f(x)=f(x+2) for all x\in \mathbb{R}. Equipping V with the inner product of the space C[-1,1], is the operator D:f\mapsto ... • 1,420 0 votes 0 answers 10 views ### How to represent a point's velocity seen from a moving coordinate system? let us consider a three-dimensional point x that is fixed in the world coordinate system, ie. its velocity is zero, \dot{x}=0. Consider another coordinate system that is defined by a rotation ... • 21 1 vote 0 answers 60 views ### Computing the adjoint of -\Delta In B. Helffer's Spectral Theory and its Application, Remark 2.7 p. 16 the author is considering the two following operators T_0=-\Delta with D(T_0)=C^{\infty}_{c}(\mathbb{R}^N) T_1=-\Delta ... • 555 0 votes 0 answers 37 views ### Reconstruction of an operator given the eigenfunctions and eigenvalues I am interested in operator theory, in particular if I know the sequence of eigenvalues \{\lambda_n\}_{n=1}^\infty \subset \mathbb{R} and eigenfunctions f_n \subset X of a self-adjoint ... 3 votes 1 answer 55 views ### How to Verify the Conjugate transpose The operator A is defined by the matrix$$A_B:=\begin{pmatrix} 1 & 1 & 3 \\ 0 & 5 & -1 \\ 2 & 7 & 3 \end{pmatrix} $$in the basis b_1=(1,2,1), b_2=(1,1,2), b_3=(1,1,0) of ... • 143 0 votes 1 answer 42 views ### (T^{*})^{+} = (T^{+})^{*} I'm trying to prove why (T^{*})^{+} = (T^{+})^{*} only using properties of matrix operations (I'm considering ()^{*} and ()^{+} operations as well, just to be clear). However, I assume I cannot ... 1 vote 2 answers 38 views ### How should we characterize the relationship between two matrix representations of a linear operator with respect to two different orthonormal bases? Nielsen / Chuang remark on page 71 of "Quantum Computation and Quantum Information" that, if | v_i \rangle and |w_i \rangle are orthonormal bases, then the operator U defined by \sum_{... • 51 4 votes 2 answers 80 views ### If U is a unitary linear operator, how can I show that any matrix representation of U must be a unitary matrix? Nielsen / Chuang "Quantum Computation and Quantum Information" states on p. 70: "A matrix U is said to be unitary if U^\dagger U = I. Similarly, an operator U is unitary if U^\... • 51 1 vote 1 answer 29 views ### Can be use u-substitution for calculating the adjoint of an operator in Schwartz space? I only have seen that for calculating the adjoint of an operator in \mathcal S, it used integration by parts, but I was thinking that if one can use substitution to find the adjoint. For exmple, for ... • 495 2 votes 1 answer 50 views ### S \in \mathcal{L}(L^1(\Omega)), find T^* \in \mathcal{L}(L^\infty(\Omega)) with T^*g = Sg \forall g \in L^1(\Omega) \cap L^\infty(\Omega) Below I will bring a passage from Heat Kernels by Wolfgang Arendt (Theorem 4.3.3, page 52). I need to understand it and write a more verbose report based on the chapter, however I am stuck at this ... 0 votes 0 answers 26 views ### What guarantees that the adjoint of a suitable integral operator, e.g. a Hilbert-Schmidt operator, is again an integral operator with a kernel? This is likely a silly question, but I was wondering if T is some nice integral transform, e.g. a Hilbert-Schmidt integral operator, with an, say, L^2(\mathbb{R}^n) kernel, what then guarantees ... • 1,201 2 votes 0 answers 63 views ### T^{**} = T if the space is reflexive Assuming X is reflexive, that is the linear map J_X:X \mapsto X^{**}, J_X(x)(f) = f(x) for all f \in X^* is bijective. Is it true that J_X^{-1}T^{**}J_X = T? Here's my attempt: Let x \in X,... • 338 2 votes 0 answers 19 views ### How does the boundary term appear when taking the transposed form of the inner product with a linear operator I'm trying to figure out how the second term appears here:$$\int_\Omega \mathcal{L}(u) w \;d\Omega = \int_\Omega u \mathcal{L}^*(w) \; d\Omega + \int_\Gamma \left[S^*(w) G(u) - G^*(w)S(u)\right] d\... • 1,312 0 votes 1 answer 34 views ### Untiunitary operator on a Hilbert space A bijective linear (antilinear) operator$A$on a Hilbert space$\mathcal{H}$is called unitary (untiunitrary) if$\langle A\psi |A\phi \rangle =\langle \psi |\phi \rangle$(resp.$\langle A\psi |A\...
• 751
247 views

Let $H_1,H_2$ be Hilbert spaces with inner products $\langle \cdot, \cdot \rangle_1$ and $\langle \cdot, \cdot \rangle_2$. Corresponding to every $T \in \mathcal B(H_1,H_2)$, there is a unique element ...
• 1,088
1 vote
38 views

### Adjoint operator and random variable

Let $(\Omega,\mathcal{A},\mathbb{P})$ be a probability space and $\mathbb{D}$ a dense subset of $L^2(\Omega,\mathcal{A},\mathbb{P})$. I consider a linear map $D$ from $\mathbb{D}\subset L^2(\Omega)$ ...
• 1,367
28 views

• 71