The projection constant $Π(E):=Π(E, \ell_\infty)$ of a finite-dimensional Banach space $E\subset\ell_\infty$ is by definition the smallest norm of a linear projection of $\ell_\infty$ onto $E$. Fix $n\geq 1$ and denote by $Π_n$ the maximal value of $Π(\cdot)$ amongst $n$-dimensional real Banach spaces. We prove for every $\varepsilon >0$ that there exist an integer $d\geq 1$ and an $n$-dimensional subspace $E\subset\ell_1^d$ such that $Π_n \leq Π(E, \ell_1^d) +2 \varepsilon$ and the orthogonal projection $P\colon \ell_1^d\to E$ is almost minimal in the sense that $\lVert P \rVert \leq Π(E, \ell_1^d)+\varepsilon$. As a consequence of our main result, we obtain a formula relating $Π_n$ to smallest absolute value row-sums of orthogonal projection matrices of rank $n$.
Let u be a hermitian involution, and e an orthogonal projection, acting on the same Hilbert space. We establish the exact formula, in terms of the norm of eue, for the distance from e to the set of all orthogonal projections q from the algebra generated by e,u, and such that quq=0.
We fully characterize orthogonal projections of infinite right circular (round) cones in real Hilbert spaces. Another interpretation is that, given two vectors in a real Hilbert space, we establish the optimal estimate on the angle between the orthogonal projections of the two vectors. The estimate depends on the angle between the two vectors and the position of only one of the two vectors. Our results also make a contributions to Cauchy-Bunyakovsky-Schwarz type inequalities.
We characterize the sets $\XX$ of all products $PQ$, and $\YY$ of all products $PQP$, where $P,Q$ run over all orthogonal projections and we solve the problems $\arg\min\{\|P-Q\|: (P,Q) \in \cal Z\}$, for $\cal Z=\XX$ or $\YY.$ We also determine the polar decompositions and Moore-Penrose pseudoinverses of elements of $\XX.$
This paper is the written version of our talk (presented by the second author) at the IWOTA in Chemnitz in August 2017. The meta theorem of the paper is that Halmos' two projections theorem is something like Robert Sheckley's Answerer: no question about the W*- and C*-algebras generated by two orthogonal projections will go unanswered, provided the question is not foolish. An alternative approach to questions about two orthogonal projections makes use of the supersymmetry equality introduced by Avron, Seiler, and Simon. A noteworthy insight of the paper reveals that the supersymmetric approach…
Let $P$ and $Q$ be two orthogonal projections on a separable Hilbert space, $\calH$. Wang, Du and Dou proved that there exists a unitary, $U$, with $UPU^{-1} =Q, \quad UQU^{-1} = P$ if and only if $\dim(\ker P \cap \ker(1-Q)) = \dim(\ker Q \cap \ker(1-P))$ (both may be infinite). We provide a new proof using the supersymmetric machinery of Avron, Seiler and Simon.
The aim of this paper is to describe the closure of the numerical range of the product of two orthogonal projections in Hilbert space as a closed convex hull of some explicit ellipses parametrized by points in the spectrum. Several improvements (removing the closure of the numerical range of the operator, using a parametrization after its eigenvalues) are possible under additional assumptions. An estimate of the least angular opening of a sector with vertex 1 containing the numerical range of a product of two orthogonal projections onto two subspaces is given in terms of the cosine of the Frie…
In this note, for any two orthogonal projection $P,Q$ on a Hilbert space, the characterization of spectrum of anticommutator $PQ+QP$ has been obtained. As a corollary, the norm formula $$\parallel PQ+QP\parallel=\parallel PQ\parallel+\parallel PQ\parallel^2$$ has been got an alternative proof (see Sam Waltrs, Anticommutator norm formula for projection operators, arXiv:1604.00699vl [math.FA] 3 Apr 2016)
We construct a single smooth orthogonal projection with desired localization whose average under a group action yields the decomposition of the identity operator. For any full rank lattice $Γ\subset\mathbb R^d$, a smooth projection is localized in a neighborhood of an arbitrary precompact fundamental domain $\mathbb R^d/Γ$. We also show the existence of a highly localized smooth orthogonal projection, whose Marcinkiewicz average under the action of $SO(d)$, is a multiple of the identity on $L^2(\mathbb S^{d-1})$. As an application we construct highly localized continuous Parseval frames on the…
The main result of this paper refers to the boundedness of the orthogonal projection $P_α:L^{2}(\mathbb{R}^{n},dμ_α)\rightarrow \mathcal{H}_α^{2}, n\geq2 $ associated to the harmonic Fock space $\mathcal{H}_α^{2},$ where $dμ_α(x)=(πα)^{-n/2}e^{-\frac{|x|^2}α}dx.$ We prove that the operator $P_α$ is not bounded on $L^{p}(\mathbb{R}^{n},dμ_β)$ when $0<p< 1$ and we found a necessary and sufficient condition for the boundedness when $1\leq p<\infty$ and $n$ is an even integer.
We describe absolutely ordered $p$-normed spaces, for $1 \le p \le \infty$ which presents a model for "non-commutative" vector lattices and includes order theoretic orthogonality. To demonstrate its relevance, we introduce the notion of {\it absolute compatibility} among positive elements in absolute order unit spaces and relate it to symmetrized product in the case of a C$^{\ast}$-algebra. In the latter case, whenever one of the elements is a projection, the elements are absolutely compatible if and only if they commute. We develop an order theoretic prototype of the results. For this purpose…
The main result of this paper is a proof that, for any $f \in L_1[a,b]$, a sequence of its orthogonal projections $(P_{Δ_n}(f))$ onto splines of order $k$ with arbitrary knots $Δ_n$, converges almost everywhere provided that the mesh diameter $|Δ_n|$ tends to zero, namely \[ f \in L_1[a,b] \Rightarrow P_{Δ_n}(f,x) \to f(x) \quad \mbox{a.e.} \quad (|Δ_n|\to 0)\,. \] This extends the earlier result that, for $f \in L_p$, we have convergence $P_{Δ_n}(f) \to f$ in the $L_p$-norm for $1 \le p \le \infty$.}
Projections are constructed in the rotation algebra that are orthogonal to their Fourier transform and which are fixed under the flip automorphism. Such projections are expected in a construction of an inductive limit structure for the irrational rotation algebra that is invariant under the Fourier transform. (Namely, as two circle algebras of the same dimension, that are swapped by the Fourier transform, plus a few points.) The calculation is based on Rieffel's construction of the Schwartz space as an equivalence bimodule of rotation algebras as well as on the theory of Theta functions.
In this paper, based on the block operator technique and operator spectral theory, the general explicit expressions for intertwining operators and direct rotations of two orthogonal projections have been established. As a consequence, it is an improvement of Kato's result (Perturbation Theory of Linear operators, Springer-Verlag, Berlin/Heidelberg, 1996); J. Avron, R. Seiler and B. Simon's Theorem 2.3 (The index of a pair of projections, J. Funct. Anal. 120(1994) 220-237) and C. Davis, W.M. Kahan, (The rotation of eigenvectors by a perturbation, III. SIAM J. Numer. Anal. 7(1970) 1-46).
We prove that the set of orthogonal projections on a Hilbert space equipped with the length metric is $\frac\pi2$-geodesic. As an application, we consider the problem of variation of spectral subspaces for bounded linear self-adjoint operators and obtain a new estimate on the norm of the difference of two spectral projections associated with isolated parts of the spectrum of the perturbed and unpertubed operators, respectively. In particular, recent results by Kostrykin, Makarov and Motovilov from [Trans. Amer. Math. Soc., V. 359, No. 1, 77 -- 89] and [Proc. Amer. Math. Soc., 131, 3469 -- 3476…
We consider the Haar functions $h_I$ on dyadic intervals. We show that if $p>\frac23$ and $E\subset[0,1]$ then the set of all functions $\|h_I1_E\|_2^{-1}h_I1_E$ with $|I\cap E|\geq p|I|$ is a Riesz sequence. For $p\leq\frac23$ we provide a counterexample.
In the article is shown that classes of finite equivalence that are generated by Lebesgue-Riesz spaces Lp have the amalgamation property if and only if p is not equal to an even natural number that is strongly large then 2.
We prove large deviation principles (LDPs) for random matrices in the orthogonal group and Stiefel manifold, determining both the speed and good convex rate functions that are explicitly given in terms of certain log-determinants of trace-class operators and are finite on the set of Hilbert-Schmidt operators $M$ satisfying $\|MM^*\|<1$. As an application of those LDPs, we determine the precise large deviation behavior of $k$-dimensional random projections of high-dimensional product distributions using an appropriate interpretation in terms of point processes, also characterizing the space of …
The notion of quasi-unit has been introduced by Yosida in unital Riesz spaces. Later on, a fruitful potential theoretic generalization was obtained by Arsove and Leutwiler. Due to the work of Eriksson and Leutwiler, this notion also turned out to be an effective tool by investigating the extreme structure of operator segments. This paper has multiple purposes which are interwoven, and are intended to be equally important. On the one hand, we identify quasi-units as orthogonal projections acting on an appropriate auxiliary Hilbert space. As projections form a lattice and are extremal points of …
Let $d\in\mathbb N$ and $f$ be a function in the Orlicz class $L(\log^+L)^{d-1}$ defined on the unit cube $[0,1]^d$ in $\mathbb{R}^d$. Given partitions $Δ_1,\ldots,$ $Δ_d$ of $[0,1]$, we first prove that the orthogonal projection $P_{(Δ_1,\dots,Δ_d)}(f)$ onto the space of tensor product splines with arbitrary orders $(k_1,\dots, k_d)$ and knots $Δ_1,\ldots,Δ_d$ converges to $f$ almost everywhere as the mesh diameters $|Δ_1|,\ldots, |Δ_{d}|$ tend to zero. This extends the one-dimensional result in [Passenbrunner and Shadrin, Journal of Approximation Theory, 2014] to arbitrary dimensions. In a s…
We consider the problem of $L^1$ (un)boundedness for a wide class of orthogonal projections, including Bergman projections on domains in complex manifolds and Szegö projections on abstract CR manifolds.
