Questions tagged [hilbert-spaces]
A Hilbert space $H$ is a real or complex vector space endowed with an inner product such that $H$ is a complete metric space when endowed with the norm induced by this inner product.
2
votes
1answer
63 views
volume of parallelotope in $L^2(\mathbb R).$
Let $L^2(\mathbb R)$ is complex Hilbert space with standard inner product.
Does it make sense to talk of volume of parallelotope formed by following vectors in $L^2(\mathbb R):$ say, e.g.,
$$\{ f(...
4
votes
0answers
38 views
Interpolation of some Sobolev spaces
Let $X_0=L^2(0,1)$, $X_1=H^4(0,1)$, $X_2=H^4(0,1)\cap H^2_0(0,1)$. We know the interpolation space $$(X_0,X_1)_{1/2,2}=H^2(0,1).$$
I am wondering what is
$$(X_0,X_2)_{1/2,2}=?$$
Would it be $H^2_0(0,...
0
votes
2answers
99 views
Spectrum equals eigenvalues for unbounded operator
Let $D$ be an unbounded densely defined operator on a separable Hilbert space $H$. If $D$ is diagonalisable with all eigenvalues having finite multiplicity and growing towards infinity, does it follow ...
7
votes
1answer
152 views
Almost orthonormal projection and orthonormal projection in Hilbert space
Let $(e_i)_i$ be a family of vectors in a Hilbert space being almost orthonormal but not quite, i.e.
$$\langle e_i, e_j \rangle \approx \delta_{i,j} + \alpha e^{-\vert i-j \vert} $$
and $\alpha$ is ...
1
vote
0answers
78 views
Limit of sequence of vectors in $\ell^2$ with coefficients approaching $0$
Let $\{v_m\}_{m \in \mathbb{N}} \subset \ell^2$ be a sequence in $\ell^2$ over the complex plane $\mathbb{C}$ such that: $\{v_m\}_{m \in \mathbb{N}}$ is linearly independend and $v_m \to v$
Let $V= \...
3
votes
0answers
63 views
Structure of a group acting on a Hilbert space
Assume a group $G$ acts faithfully by isometries on a separable infinite dimensional Hilbert space $H$ in such a way that the orbits are closed and the quotient $H/G$ is isometric to a finite ...
6
votes
1answer
126 views
If $\ $ $yx_n\to 0 $ for all $y$ in a C$^*$-algebra, Is it true that $x_n$ is weakly convergent to $0$?
$A$ is a C$^*\! $-algebra and $(x_n)_{n\in \mathbb{N}} \subseteq A $.
If $\ $ $yx_n\to 0 $ for all $y\in A$, Is it true that $x_n$ is weakly
convergent to $0$ ?
For unitals this is trivial. ...
4
votes
0answers
65 views
Orthonormal Basis of Multi-Dimensional Sobolev Space of Different Orders without Reproducing Kernel
Let $\Omega$ be an open subset of $\mathbb{R}^d$. Under regularity conditions, we know that the $s$-th order Sobolev space $H^s(\Omega)$ with $s \geq d/2$ is a reproducing kernel Hilbert space. In ...
8
votes
1answer
241 views
What's so special about the Orthonormal base $\{e_n\}$ of $L^2[0,1]$, where $e_n(x)=e^{2\pi i nx }$?
Let $f \in L^2([0,1])$ . Then Carleson's Theorem states that
$$\lim_{N\to \infty} \sum_{|n|<N} \langle f,e_n\rangle e_n(x)=f(x),\quad\text{a.e. } x\in[0,1],$$
where $\{e_n\}$ is the Orthonormal ...
0
votes
1answer
103 views
Smallest collection of linear operators satisfying isometry property
Let $\mathscr{A}=\{(A_{1,1},A_{1,2},A_{1,3}),...,(A_{S,1},A_{S,2},A_{S,3})\}$ be a collection of linear operators $A_{n,k}:\mathbb{R}^2 \rightarrow \mathbb{R}^2$. For $u$ and $v\in (\mathbb{R}^2)^3$, ...
3
votes
1answer
86 views
Hoeffding's inequality for Hilbert space valued random elements
Suppose that $\mathbb H$ is a separable Hilbert space and $X_1,\ldots,X_n$ are independent zero mean $\mathbb H$-valued random elements such that $\|X_i\|\le s$ for each $1\le i\le n$, where $\|\cdot\|...
4
votes
2answers
152 views
Compact images of nowhere dense closed convex sets in a Hilbert space
Let $B=-B$ be a nowhere dense bounded closed convex set in the Hilbert space $\ell_2$ such that the linear hull of $B$ is dense in $\ell_2$.
Question. Is there a non-compact linear bounded operator ...
2
votes
1answer
127 views
Strongly Continuous Group Actions on the $ C^{\ast} $-Algebra of Compact Operators on a Hilbert Space
Let $ \mathcal{H} $ be a not-necessarily-separable Hilbert space. Let $ G $ be a locally compact Hausdorff group. It is easy to see that if $ U: G \to \mathbb{U}(\mathcal{H}) $ is a norm-continuous ...
3
votes
1answer
115 views
A specific problem on : Can bounding the Sobolev norm, bound a higher derivative?
Let $f \in H^k(\mathbb{R}^m)$, $k>\frac{m}{2}$. Given any $f$, such that $\|f\|_{H^k(\mathbb{R}^m)}<K$ , and any $\phi \in C^{\infty}(\mathbb{R}^m)\cap H^k(\mathbb{R}^m)$, such that $\|\phi\|_{...
2
votes
1answer
48 views
Are the intersection of proximinal sets in a Hilbert Space proximinal?
Let $X$ be a real Hilbert Space and $C \subseteq X$. Let $d_C$ be the infimal distance function to $C$ and $P_C(x) = C \cap S[x; d_C(x)]$ be the metric projection. We say $C$ is proximinal if $P_C(x) \...
-2
votes
1answer
135 views
Weak convergent $+$ strongly convergent subsequence $\Rightarrow$ strong convergence? [closed]
Let $X$ be a Hilbert space containing functions defined over a bounded region $\Omega\subset \mathbb{R}^N$. Assume $f_n\in X$ converges weakly to $f\in X$, and also has a strongly convergent ...
2
votes
0answers
44 views
Is every nonexpansive retract of a Hilbert space closed and convex?
Given a closed and convex subset $C\subset H$ of a Hilbert space $H$, the metric projection is a nonexpansive retraction of $H$ onto $C$. This implies that every closed and convex subset of a Hilbert ...
14
votes
2answers
458 views
Multiple of identity plus compact
Is there an example of a bounded operator $T\in\mathcal{B}(H)$, where $H$ is a separable complex Hilbert space, such that no restriction to an infinite dimensional closed subspace is multiple of ...
4
votes
1answer
84 views
Non-point spectrum for diagonalisable self-adjoint unbounded operator
Given a (separable) Hilbert space H and an unbounded densely defined linear operator $T:{\cal D}(T) \to $H such that ${\cal D}$ is diagonalizable (it means $\exists$ an O.N.B. of H such that all basis ...
5
votes
1answer
131 views
Hilbert representation of a bilinear form
Let $\sigma:\mathbb{R}^n\times \mathbb{R}^n\to \mathbb{R}$ be a bilinear symmetric form which is non-degenerate in the sense that for every $0\neq u\in \mathbb{R}^n$ there is $v\in \mathbb{R}^n$ with $...
5
votes
1answer
132 views
Unbounded version of continuous functional calculus
For a normal operator $T$ on a Hilbert space ${\cal H}$, it is well known that for any continuous complex valued function $f$ on the spectrum of $T$, we have a well-defined operator $f(T) \in B({\cal ...
1
vote
1answer
90 views
Optimal estimate in trace norm
Let $x,y$ be vectors of some Hilbert space of unit length.
Then we can consider the projection $P_x:=\langle \bullet, x \rangle x$ and similarly $P_y.$
Assume then that we know that $\left\lVert x-...
3
votes
0answers
71 views
Hilbert space separability for spectral triples
A spectral triple $({\cal A},{\cal H},D)$ consists of a unital $*$-algebra ${\cal A}$ represented as bounded operators on a Hilbert space ${\cal H}$, together with an unbounded operator $D$ having ...
1
vote
0answers
128 views
Why is $H^{1/2}$ a Hilbert space?
Let $n\in\mathbb{N}$ and $\Omega \subseteq \mathbb{R}^n$ sufficiently smooth. Then we have the Hilbert space $H^1(\Omega)$ and the trace operator $\operatorname{tr}: H^1(\Omega) \to L^2(\partial \...
2
votes
1answer
82 views
Fredholmness of formal selfadjoint operator $AA^*$ and Fredholmenss of $A$
Let $X$ and $Y$ be Hilbert spaces with respective inner products $\langle , \rangle_{X,Y}$. Let $A:X \rightarrow Y$ be a bounded linear operator. Assume there is a non-degenerate sesquilinear product $...
3
votes
0answers
46 views
A question about a theorem in 'Quantum dynamical semigroups generated by noncommutative unbounded elliptic operators'
I have asked this question on stack and someone advised me to ask it here. The link is https://math.stackexchange.com/questions/2900658/a-question-about-a-theorem-in-quantum-dynamical-semigroups-...
3
votes
1answer
81 views
Measurability of specific function
Let $I\subset\mathbb{R}$ denote an open and bounded interval of the real line, $H_0^1(I)$ all quadratic integrable Sobolev functions and $C(\bar{I})$ all continuous functions on said interval.
Since ...
1
vote
1answer
84 views
Orthogonal complement vector space
Let $X$ be a vector space contained in $H^{1}(\mathbb R^d),$ then we can study
$X^{\perp_{L^2}}:=\left\{ \xi \in L^2; \langle \xi, x \rangle_{L^2} =0 \ \forall x \in X \right\}$
and
$X^{\perp_{H^{-...
1
vote
0answers
73 views
Injective differential of linear operators on a Hilbertspace
Given a complex Hilbertspace $\mathcal{H}$ of dimension $\dim(\mathcal{H}) = d$ and the set $$\mathcal{F} := \{q\in L(\mathcal{H})\vert\quad \text{rank}(q) = 4 \quad \wedge \lambda^q_{1,2} < 0\ \ \...
1
vote
0answers
879 views
A Question on certain Hilbert space of continuous functions, and a characteristic of convergence in it
Define $T^k(\Omega)$, $\Omega$ an open subset of $\mathbb{R}^m$ (with a smooth boundary), as a space of function equivalance classes, with the norm defined as $$ \|f\|_{T^k(\Omega)}^2 = \|f\|_{L^2(...
3
votes
0answers
56 views
Constant in trace theorem for balls
Consider the standard open ball $B_r:=\left\{x ; \left\lvert x \right\rvert \le R \right\}.$
The trace theorem tells us any function in $W^{k,p}(B_r)$ can be restricted to a function $W^{k-1,p}(\...
1
vote
1answer
115 views
The tensor product of two bounded operators
Let $E$, $F$ be two complex Hilbert spaces and $\mathcal{L}(E)$ (resp. $\mathcal{L}(F)$) be the algebra of all bounded linear operators on $E$ (resp. $F$).
The algebraic tensor product of $E$ and $F$ ...
3
votes
1answer
276 views
What is the consistency strength of non-existence of outer automorphisms of Calkin algebra?
The Calkin algebra $C(H)$ is the quotient of $B(H)$, the ring of bounded linear operators on a separable infinite-dimensional Hilbert space $H$, by the ideal $K(H)$ of compact operators.
In 1977, ...
5
votes
1answer
213 views
Fell's trick for Lie groups
Let $\Gamma$ be a countable discrete group and $\lambda$ the left regular representation of $\Gamma$ on $l^2(\Gamma)$. Let $\rho:\Gamma\rightarrow U(H)$ be a unitary representation of $\Gamma$ on some ...
1
vote
1answer
53 views
Completeness of Lowner order in separable Hilbert space
Consider a separable Hilbert space $\mathcal{H}$, we can first define $T(\mathcal{H})$ the trace class of $\mathcal{H}$, then $D(\mathcal{H})$ to denote the interesection of positive operator and ...
3
votes
1answer
94 views
Completely Positive Maps and their dual in Separable Hilbert Space
Consider a separable Hilbert space $\mathcal{H}$, we can first define $T(\mathcal{H})$ the trace class of $\mathcal{H}$, then $D(\mathcal{H})$ to denote the set of positive operator with trace less ...
10
votes
2answers
357 views
$x f'$ bounded by $x^2f $ and $f''$?
Consider the Hilbert space of functions $f \in L^2(\mathbb R)$ such that $x^2f \in L^2(\mathbb R) $ and $ f'' \in L^2(\mathbb R).$
I am wondering whether it is true that $xf'\in L^2(\mathbb R)$ as ...
1
vote
0answers
75 views
“Barrier functions” in function spaces [closed]
In general the idea of a "barrier function" (as in the context of "interior point methods") can possibly be thought of as defining a function corresponding to an open subset of $\mathbb{R}^n$ such ...
2
votes
1answer
101 views
Insights about a frame-like inequality
I'm a graduate student doing research on time-frequency analysis. I am considering the existence of a certain frame-like inequality. Let $H: L^2(\mathbb{R}^d) \rightarrow L^2(\mathbb{R}^d)$ be a ...
4
votes
1answer
131 views
Extending maps from dense $*$-algebras of $C^*$-algebras
Given $\cal{A},\cal{B}$ two dense $*$-algebras of two $C^*$-algebras $A$ and $B$ respectively, together with a $*$-algebra homomorphism $f:\cal{A} \to \cal{B}$, is it clear that $f$ extends to a ...
0
votes
0answers
41 views
Uniform convergence in Hadamard derivatives
Let $T\colon X \to Y$ be a nonlinear operator between Hilbert spaces which is Lipschitz and is Hadamard differentiable. It satisfies
$$T(x+th)=T(x) + tT'(x)(h) + r(t)$$
where $r(t)=r(t,x,h)$ is the ...
3
votes
1answer
174 views
Convergence of nuclear operators
Let $H$ be a separable infinite-dimensional real Hilbert space. We consider operators in $H.$
Nuclear norm of a nuclear operator is the sum of its singular values.
A nuclear, positive and self-...
0
votes
1answer
69 views
Convergence rate of eigenvectors
Let us suppose that $A,A_1,A_2,\ldots$ are non-negative definite self-adjoint bounded linear operators in $L(\mathbb H)$, where $\mathbb H$ is a separable Hilbert space. $(v_j)_{j\ge1}$ and $(\...
7
votes
1answer
193 views
Subspaces isomorphic with quotients
Suppose $X$ is a Banach space not isomorphic to a Hilbert space. Can we always find a subspace of $X$ that is not isomorphic to a quotient of $X$?
1
vote
0answers
68 views
A nested sequence of closed subspaces of $\ell^2$
Let $\{v_n\}_{n \in \mathbb{N}} \subset \ell^2$ be a sequence in $\ell^2$ over $\mathbb{C}$ such that $\{v_n\}_{n \in \mathbb{N}}$ is linearly independent and $v_n \to u$.
Is it possible extract a ...
1
vote
1answer
131 views
On projection theory for inseparable Hilbert spaces
How can one see that $I$ is an infinite projection in $B(\mathcal{H})$, where $\mathcal{H}$ is an inseparable Hilbert space?
3
votes
0answers
143 views
Araki's proof of simple connectedness of the restricted orthogonal group
I am trying to understand Araki's proof of the statement that the restricted orthogonal group of a Hilbert space with a unitary structure is simply connected. This proof starts on page 114 of these ...
2
votes
2answers
163 views
Point spectrum of a positive invertible operator
Let $G$ be a l.c. group and $f$ belong to $C_c(G)$, the space of continuous functions with compact support. Define an operator$T_f$ on $L^2(G)$ by $T_f(g)=f*g$ (the convolution product). If $T_f$ is ...
5
votes
1answer
156 views
The largest topological copy of a Hilbert space contained in $\ell^1$
Let us consider $\ell^1$, the space of absolutely summable sequences in the space of complex numbers. Clearly every finite dimensional Hilbert space is topologically embedded into $\ell^1$.
...
2
votes
0answers
81 views
A specific Schwartz function $f$ on $\mathbb C^2$
Choose a Schwartz function on $\mathbb C$ of the form $f(z)=f(r e^{i\theta})= f_0(r) e^{in\theta}$. Then $$(*) \quad f(e^{i\alpha} z)= e^{in\alpha} f(z), \quad \forall z\in \mathbb C.$$
Now, let $f$ ...
