All Questions
Tagged with lattice-orders functional-analysis
20
questions
1
vote
1
answer
61
views
Stone's theorem in the presence of superselection rules
Let $\mathcal L$ be a orthomodular sub-$\sigma$-lattice of the lattice ${\rm L}(H)$ consisting of all closed subspaces of the separable Hilbert space $H$,
(precisely
$\mathcal L$ is a set of closed ...
- 319
0
votes
1
answer
51
views
Notation confusion for "closed linear span" and "intersection of orthogonal projections"
This is merely about notations. In Conway's Functional Analysis textbook, exercise I.2.4 on p.11 to be precise, the closed linear span of a set $A\subset H$, where $H$ is a hilbert space, is defined ...
- 1,670
1
vote
2
answers
88
views
In a vector lattice $V$, is it the case that $\sup(0,v) = P_{V^+}v$? [closed]
We have a reflexive Banach space which is strictly convex. Let $V$ be a vector lattice with a partial ordering and write $V^+ := \{v \in V : v \geq 0\}$ to be the positive cone. Suppose that it is a ...
- 1,424
3
votes
0
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38
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Regularization of $\max(0,\cdot)$ as an operator in a Hilbert space
Let $H$ be a vector lattice for a (partial) ordering $\leq$. Hence $\max(a,b)$ is defined for $a, b \in H$. Where can I find theory regarding the regularization of $\max(0,\cdot)$ as an operator? By ...
- 73
0
votes
0
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57
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General form of elements in the vector lattice (Riesz Space) generated by a vector space
Can any one tell me how the elements of the set $A^{∧∨}$ will look like? I believe they are of the following form:
$∨^m_i∧^{m_i}_{k_i} x_{i,{k_i}}$ where $x_{i,{k_i}}$ is in $A$. But in this case I'm ...
- 1,544
0
votes
1
answer
209
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Positiveness of Inverse Of Positive Operator Implies Lattice Isomorphism?
Let $X$ be a Banach Lattice and denote by $\mathcal{B}(X)$ the banach space of bounded linear endomorphisms. An operator $T \in \mathcal{B}(X)$ is called positive if $Tx \geq 0$ whenever $x \geq 0.$ ...
- 983
1
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1
answer
99
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Does the order induced by a self-dual cone produce a Riesz space?
Let $X$ be a Hilbert space, $K$ a closed, convex, and self-dual cone. The latter property means
$$
K = \{ y\in X : \ \langle x,y\rangle \ge0 \ \forall x\in K\}.
$$
Then $K$ induces an order on $X$ by
$...
- 50.4k
0
votes
1
answer
55
views
closed lattice ideal is isomorphic to $C(K)$
Let $X\in E$, where $E$ is a Banach function space on $(0,1)$. Consider the interval $[-X,X]$ and generate it to a closed lattice ideal $I$ of $E$. We may renorm this ideal $I$ such that $[-X,X]$ is ...
- 1,348
1
vote
0
answers
94
views
Establish Archimedean property of a vector-lattice
I am trying to find ways to prove the Archimedian property of a certain vector lattice and got stuck on the following type of problem.
I feel the statement below (or in fact weaker versions) should ...
- 11
2
votes
1
answer
64
views
Atomicity of blocks in a Hilbert lattice
Where can I find the proof that any block (maximal boolean subalgebra) $\mathbf{B}$ of the orthomodular lattice $\mathcal{L}$ of closed subspaces of a separable Hilbert space $\mathcal{H}$ is atomic?
- 91
4
votes
2
answers
144
views
Distributivity in boolean subalgebras of orthomodular lattice
A boolean subalgebra $B$ of the orthomodular lattice $L$ of closed subspaces of a separable Hilbert space, may be defined like a sublattice with $0$ and $1$, with pairwise commuting elements.
How to ...
- 91
5
votes
2
answers
769
views
The set of self-adjoint operators over a Hilbert space doesn't form a lattice
How can someone prove, that over a Hilbert space $\mathcal{H}$ with $dim \mathcal{H} \geq 2$, the set of all self-adjoint operators doesn't form a lattice?
I found a result from Kadison, that if two ...
- 186
1
vote
1
answer
96
views
Structure of vector topologies and locally convex topologies on a fixed vector space
Let $X$ be a real or complex vector space and consider the partially ordered sets $lc(X) \subseteq v(X) \subseteq t(X)$ of respectively locally convex topologies, vector topologies and all topologies ...
- 3,565
2
votes
1
answer
207
views
Are lattice operations in set of orthogonal projections in Hilbert space continous?
Let $H$ be Hilbert space and denote set of all orthogonal projections in $H$ by $\Pi$. Then $\Pi$ can be given structure of a lattice. We partially order it by declaring $P \leq Q$ if $Q-P$ is ...
- 3,100
0
votes
0
answers
38
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inverse relation in a vector lattice
Let $X$ be a vector lattice endowed with an ordering $\leq$. If for some operator $\phi:X_+\rightarrow X$ and $m,M\in X$, we have
$$m\leq \phi(x)\leq M,\quad x\in X_+. $$
What would be the bound for ...
- 273