Noncommutative Multivariable Operator Theory

In this paper, we study noncommutative domains ${\mathbb{D}_f^\varphi(\mathcal{H}) \subset B(\mathcal{H})^n}$ generated by positive regular free holomorphic functions f and certain classes of n-tuples ${\varphi = (\varphi_1, \ldots, \varphi_n)}$ of formal power series in noncommutative indeterminates Z ₁, . . . , Z _n . Noncommutative Poisson transforms are employed to show that each abstract domain ${\mathbb{D}_f^\varphi}$ has a universal model consisting of multiplication operators (M _Z1, . . . , M _{Z n} ) acting on a Hilbert space of formal power series. We provide a Beurling type characterization of all joint invariant subspaces under M _Z1, . . . , M _{Z n} and show that all pure n-tuples of operators in ${\mathbb{D}_f^\varphi(\mathcal{H})}$ are compressions of ${M_{Z_1} \otimes I, \ldots, M_{Z_n} \otimes I}$ to their coinvariant subspaces. We show that the eigenvectors of ${M_{Z_1}^*, \ldots, M_{Z_n}^*}$ are precisely the noncommutative Poisson kernels ${\Gamma_\lambda}$ associated with the elements ${\lambda}$ of the scalar domain ${\mathbb{D}_{f,<}^\varphi(\mathbb{C}) \subset \mathbb{C}^n}$ . These are used to solve the Nevanlinna-Pick interpolation problem for the noncommutative Hardy algebra ${H^\infty(\mathbb{D}_f^\varphi)}$ . We introduce the characteristic function of an n-tuple ${T=(T_1, \ldots , T_n) \in \mathbb{D}_f^\varphi(\mathcal{H})}$ , present a model for pure n-tuples of operators in the noncommutative domain ${\mathbb{D}_f^\varphi(\mathcal{H})}$ in terms of characteristic functions, and show that the characteristic function is a complete unitary invariant for pure n-tuples of operators in ${\mathbb{D}_f^\varphi(\mathcal{H})}$ .     

On a problem concerning the Banach algebra generated by the maps n\mapsto\lambda^{\binom{n}{k}}

In Jabbari and Namioka (Milan J. Math. 78:503?C522, 2010), the authors characterized the spectrum M(W) of the Weyl algebra W, i.e. the norm closure of the algebra generated by the family of functions $\{n\mapsto x^{n^{k}}; x\in\mathbb{T}, k\in\mathbb{N}\}$ , ( $\mathbb{T}$ the unit circle), with a closed subgroup of $E(\mathbb{T})^{\mathbb{N}}$ where $E(\mathbb{T})$ denotes the family of the endomorphisms of the multiplicative group $\mathbb{T}$ . But the size of M(W) in $E(\mathbb{T})^{\mathbb{N}}$ as well as the induced group operation were left as a problem. In this paper, we will give a solution to this problem.     

A decomposition of the universal embedding space for the near polygon {{\mathbb H}_n}

Let ${{\mathbb H}_n, n \geq 1}$ , be the near 2n-gon defined on the 1-factors of the complete graph on 2n?+?2 vertices, and let e denote the absolutely universal embedding of ${{\mathbb H}_n}$ into PG(W), where W is a ${\frac{1}{n+2} \left(\begin{array}{c}2n+2 \\ n+1\end{array}\right)}$ -dimensional vector space over the field ${{\mathbb F}_2}$ with two elements. For every point z of ${{\mathbb H}_n}$ and every ${i \in {\mathbb N}}$ , let Δ _i (z) denote the set of points of ${{\mathbb H}_n}$ at distance i from z. We show that for every pair {x, y} of mutually opposite points of ${{\mathbb H}_n, W}$ can be written as a direct sum ${W_0 \oplus W_1 \oplus \cdots \oplus W_n}$ such that the following four properties hold for every ${i \in \{0,\ldots,n \}}$ : (1) ${\langle e(\Delta_i(x) \cap \Delta_{n-i}(y)) \rangle = {\rm PG}(W_i)}$ ; (2) ${\left\langle e \left( \bigcup_{j \leq i} \Delta_j(x) \right) \right\rangle = {\rm PG}(W_0 \oplus W_1 \oplus \cdots \oplus W_i)}$ ; (3) ${\left\langle e \left( \bigcup_{j \leq i} \Delta_j(y) \right) \right\rangle = {\rm PG}(W_{n-i}\oplus W_{n-i+1} \oplus \cdots \oplus W_n)}$ ; (4) ${\dim(W_i) = |\Delta_i(x) \cap \Delta_{n-i}(y)| = \left(\begin{array}{c}n \\ i\end{array}\right)^2 - \left(\begin{array}{c}n \\ i-1\end{array}\right) \cdot \left(\begin{array}{c}n \\ i+1\end{array}\right)}$ .     

Permanent Versus Determinant over a Finite Field

Let $ \mathbb{F} $ be a finite field of characteristic different from 2. We study the cardinality of sets of matrices with a given determinant or a given permanent for the set of Hermitian matrices $ {{\mathcal{H}}_n}\left( \mathbb{F} \right) $ and for the whole matrix space M _n ( $ \mathbb{F} $ ). It is known that for n = 2, there are bijective linear maps Φ on $ {{\mathcal{H}}_n}\left( \mathbb{F} \right) $ and M _n ( $ \mathbb{F} $ ) satisfying the condition per A = det Φ(A). As an application of the obtained results, we show that if n ≥ 3, then the situation is completely different and already for n = 3, there is no pair of maps (Φ, ?), where Φ is an arbitrary bijective map on matrices and ? : $ \mathbb{F} $ → $ \mathbb{F} $ is an arbitrary map such that per A = ?(det Φ(A)) for all matrices A from the spaces $ {{\mathcal{H}}_n}\left( \mathbb{F} \right) $ and M _n ( $ \mathbb{F} $ ), respectively. Moreover, for the space M _n ( $ \mathbb{F} $ ), we show that such a pair of transformations does not exist also for an arbitrary n > 3 if the field $ \mathbb{F} $ contains sufficiently many elements (depending on n). Our results are illustrated by a number of examples.     

Tail homogeneity of invariant measures of multidimensional stochastic recursions in a critical case

We consider the stochastic recursion ${X_{n+1} = M_{n+1}X_{n} + Q_{n+1}, (n \in \mathbb{N})}$ , where ${Q_n, X_n \in \mathbb{R}^d }$ , M _n are similarities of the Euclidean space ${ \mathbb{R}^d }$ and (Q _n , M _n ) are i.i.d. We study asymptotic properties at infinity of the invariant measure for the Markov chain X _n under assumption ${\mathbb{E}{[\log|M|]}=0}$ i.e. in the so called critical case.     

On certain forms and quadrics related to symplectic dual polar spaces in characteristic 2

Let V be a 2n-dimensional vector space over a field ${\mathbb {F}}$ and ξ a non-degenerate alternating form defined on V. Let Δ be the building of type C _n formed by the totally ξ-isotropic subspaces of V and, for 1 ≤ k ≤ n, let ${\mathcal {G}_k}$ and Δ _k be the k-grassmannians of PG(V) and Δ, embedded in ${W_k=\wedge^kV}$ and in a subspace ${V_k\subseteq W_k}$ respectively, where ${{\rm dim}(V_k)={2n \choose k} - {2n \choose {k-2}}}$ . This paper is a continuation of Cardinali and Pasini (Des. Codes. Cryptogr., to appear). In Cardinali and Pasini (to appear), focusing on the case of k = n, we considered two forms α and β related to the notion of ‘being at non maximal distance’ in ${\mathcal {G}_n}$ and Δ _n and, under the hypothesis that ${{\rm char}(\mathbb {F}) \neq 2}$ , we studied the subspaces of W _n where α and β coincide or are opposite. In this paper we assume that ${{\rm char}(\mathbb {F}) = 2}$ . We determine which of the quadrics associated to α or β are preserved by the group ${G= {\rm Sp}(2n, \mathbb {F})}$ in its action on W _n and we study the subspace ${\mathcal {D}}$ of W _n formed by vectors v such that α(v, x) = β(v, x) for every ${x \in W_n}$ . Finally, we show how properties of ${\mathcal {D}}$ can be exploited to investigate the poset of G-invariant subspaces of V _k for k = n ? 2i and ${1\leq i \leq \lfloor n/2\rfloor}$ .     

Boundedness and Compactness of Pseudodifferential Operators with Non-Regular Symbols on Weighted Lebesgue Spaces

Applying the boundedness on weighted Lebesgue spaces of the maximal singular integral operator S _* related to the Carleson?CHunt theorem on almost everywhere convergence, we study the boundedness and compactness of pseudodifferential operators a(x, D) with non-regular symbols in ${L^\infty(\mathbb{R}, V(\mathbb{R})), PC(\overline{\mathbb{R}}, V(\mathbb{R}))}$ and ${\Lambda_\gamma(\mathbb{R}, V_d(\mathbb{R}))}$ on the weighted Lebesgue spaces ${L^p(\mathbb{R},w)}$ , with 1?< p <? ?? and ${w\in A_p(\mathbb{R})}$ . The Banach algebras ${L^\infty(\mathbb{R}, V(\mathbb{R}))}$ and ${PC(\overline{\mathbb{R}}, V(\mathbb{R}))}$ consist, respectively, of all bounded measurable or piecewise continuous ${V(\mathbb{R})}$ -valued functions on ${\mathbb{R}}$ where ${V(\mathbb{R})}$ is the Banach algebra of all functions on ${\mathbb{R}}$ of bounded total variation, and the Banach algebra ${\Lambda_\gamma(\mathbb{R}, V_d(\mathbb{R}))}$ consists of all Lipschitz ${V_d(\mathbb{R})}$ -valued functions of exponent ${\gamma \in (0,1]}$ on ${\mathbb{R}}$ where ${V_d(\mathbb{R})}$ is the Banach algebra of all functions on ${\mathbb{R}}$ of bounded variation on dyadic shells. Finally, for the Banach algebra ${\mathfrak{A}_{p,w}}$ generated by all pseudodifferential operators a(x, D) with symbols ${a(x, \lambda) \in PC(\overline{\mathbb{R}}, V(\mathbb{R}))}$ on the space ${L^p(\mathbb{R}, w)}$ , we construct a non-commutative Fredholm symbol calculus and give a Fredholm criterion for the operators ${A \in \mathfrak{A}_{p,w}}$ .     

Compact Hankel Operators on Generalized Bergman Spaces of the Polydisc

Let ${\vartheta}$ be a measure on the polydisc ${\mathbb{D}^n}$ which is the product of n regular Borel probability measures so that ${\vartheta([r,1)^n\times\mathbb{T}^n) >0 }$ for all 0 < r < 1. The Bergman space ${A^2_{\vartheta}}$ consists of all holomorphic functions that are square integrable with respect to ${\vartheta}$ . In one dimension, it is well known that if f is continuous on the closed disc ${\overline{\mathbb{D}}}$ , then the Hankel operator H _f is compact on ${A^2_\vartheta}$ . In this paper we show that for n ≥ 2 and f a continuous function on ${{\overline{\mathbb{D}}}^n}$ , H _f is compact on ${A^2_\vartheta}$ if and only if there is a decomposition f = h + g, where h belongs to ${A^2_\vartheta}$ and ${\lim_{z\to\partial\mathbb{D}^n}g(z)=0}$ .     

Computing α-Invariants of Singular del Pezzo Surfaces

We prove a new local inequality for divisors on surfaces and utilize it to compute α-invariants of singular del Pezzo surfaces, which implies that del Pezzo surfaces of degree one whose singular points are of type $\mathbb{A}_{1}$ , $\mathbb{A}_{2}$ , $\mathbb{A}_{3}$ , $\mathbb{A}_{4}$ , $\mathbb{A}_{5}$ , or $\mathbb{A}_{6}$ are Kähler-Einstein.     

Representation of cyclotomic fields and their subfields

Let $\mathbb{K}$ be a finite extension of a characteristic zero field $\mathbb{F}$ . We say that a pair of n × n matrices (A,B) over $\mathbb{F}$ represents $\mathbb{K}$ if $\mathbb{K} \cong {{\mathbb{F}\left[ A \right]} \mathord{\left/ {\vphantom {{\mathbb{F}\left[ A \right]} {\left\langle B \right\rangle }}} \right. \kern-0em} {\left\langle B \right\rangle }}$ , where $\mathbb{F}\left[ A \right]$ denotes the subalgebra of $\mathbb{M}_n \left( \mathbb{F} \right)$ containing A and 〈B〉 is an ideal in $\mathbb{F}\left[ A \right]$ , generated by B. In particular, A is said to represent the field $\mathbb{K}$ if there exists an irreducible polynomial $q\left( x \right) \in \mathbb{F}\left[ x \right]$ which divides the minimal polynomial of A and $\mathbb{K} \cong {{\mathbb{F}\left[ A \right]} \mathord{\left/ {\vphantom {{\mathbb{F}\left[ A \right]} {\left\langle {q\left( A \right)} \right\rangle }}} \right. \kern-0em} {\left\langle {q\left( A \right)} \right\rangle }}$ . In this paper, we identify the smallest order circulant matrix representation for any subfield of a cyclotomic field. Furthermore, if p is a prime and $\mathbb{K}$ is a subfield of the p-th cyclotomic field, then we obtain a zero-one circulant matrix A of size p × p such that (A, J) represents $\mathbb{K}$ , where J is the matrix with all entries 1. In case, the integer n has at most two distinct prime factors, we find the smallest order 0, 1-companion matrix that represents the n-th cyclotomic field. We also find bounds on the size of such companion matrices when n has more than two prime factors.     

On the Linear Dependence of a Finite Set of Additive Functions

In this note we prove the following: Let n?≥ 2 be a fixed integer. A system of additive functions ${A_{1},A_{2},\ldots,A_{n}:\mathbb{R} \to\mathbb{R}}$ is linearly dependent (as elements of the ${\mathbb{R}}$ vector space ${\mathbb{R}^{\mathbb{R}}}$ ), if and only if, there exists an indefinite quadratic form ${Q:\mathbb{R}^{n}\to\mathbb{R} }$ such that ${Q(A_{1}(x),A_{2}(x),\ldots,A_{n}(x))\geq 0}$ or ${Q(A_{1}(x),A_{2}(x),\ldots,A_{n}(x))\leq 0}$ holds for all ${x\in\mathbb{R}}$ .     

Heavy tail phenomenon and convergence to stable laws for iterated Lipschitz maps

We consider the Markov chain ${\{X_n^x\}_{n=0}^\infty}$ on ${\mathbb{R}^d}$ defined by the stochastic recursion ${X_{n}^{x}= \psi_{\theta_{n}} (X_{n-1}^{x})}$ , starting at ${x\in\mathbb{R}^d}$ , where ?? ₁, ?? ₂, . . . are i.i.d. random variables taking their values in a metric space ${(\Theta, \mathfrak{r})}$ , and ${\psi_{\theta_{n}} :\mathbb{R}^d\mapsto\mathbb{R}^d}$ are Lipschitz maps. Assume that the Markov chain has a unique stationary measure ??. Under appropriate assumptions on ${\psi_{\theta_n}}$ , we will show that the measure ?? has a heavy tail with the exponent ???>?0 i.e. ${\nu(\{x\in\mathbb{R}^d: |x| > t\})\asymp t^{-\alpha}}$ . Using this result we show that properly normalized Birkhoff sums ${S_n^x=\sum_{k=1}^n X_k^x}$ , converge in law to an ??-stable law for ${\alpha\in(0, 2]}$ .     

The Generalization of the Decomposition of Functions by Energy Operators

This work starts with the introduction of a family of differential energy operators. Energy operators $({\varPsi}_{R}^{+}, {\varPsi}_{R}^{-})$ were defined together with a method to decompose the wave equation in a previous work. Here the energy operators are defined following the order of their derivatives $(\varPsi^{-}_{k}, \varPsi^{+}_{k}, k=\{0,\pm 1,\pm 2,\ldots\})$ . The main part of the work demonstrates for any smooth real-valued function f in the Schwartz space $(\mathbf{S}^{-}(\mathbb{R}))$ , the successive derivatives of the n-th power of f ( $n \in \mathbb{Z}$ and n≠0) can be decomposed using only $\varPsi^{+}_{k}$ (Lemma); or if f in a subset of $\mathbf{S}^{-}(\mathbb{R})$ , called $\mathbf{s}^{-}(\mathbb{R})$ , $\varPsi^{+}_{k}$ and $\varPsi^{-}_{k}$ ( $k\in \mathbb{Z}$ ) decompose in a unique way the successive derivatives of the n-th power of f (Theorem). Some properties of the Kernel and the Image of the energy operators are given along with the development. Finally, the paper ends with the application to the energy function.     

On Rationality of W-algebras

We study the problem of classification of triples ( $ \mathfrak{g} $ ; f; k), where g is a simple Lie algebra, f its nilpotent element and k ∈ $ \mathbb{C} $ , for which the simple W-algebra W _k ( $ \mathfrak{g} $ ; f) is rational.     

Generalized Complex Spherical Harmonics, Frame Functions, and Gleason Theorem

Consider a finite dimensional complex Hilbert space ${\mathcal{H}}$ , with ${dim(\mathcal{H}) \geq 3}$ , define ${\mathbb{S}(\mathcal{H}):= \{x\in \mathcal{H} \:|\: \|x\|=1\}}$ , and let ${\nu_\mathcal{H}}$ be the unique regular Borel positive measure invariant under the action of the unitary operators in ${\mathcal{H}}$ , with ${\nu_\mathcal{H}(\mathbb{S}(\mathcal{H}))=1}$ . We prove that if a complex frame function ${f : \mathbb{S}(\mathcal{H})\to \mathbb{C}}$ satisfies ${f \in \mathbb{L}^2(\mathbb{S}(\mathcal{H}), \nu_\mathcal{H})}$ , then it verifies Gleason’s statement: there is a unique linear operator ${A: \mathcal{H} \to \mathcal{H}}$ such that ${f(u) = \langle u| A u\rangle}$ for every ${u \in \mathbb{S}(\mathcal{H}).\,A}$ is Hermitean when f is real. No boundedness requirement is thus assumed on f a priori.     

Saddle solutions for bistable symmetric semilinear elliptic equations

This paper concerns the existence and asymptotic characterization of saddle solutions in ${\mathbb {R}^{3}}$ for semilinear elliptic equations of the form $$-\Delta u + W'(u) = 0,\quad (x, y, z) \in {\mathbb {R}^{3}} \qquad\qquad\qquad (0.1)$$ where ${W \in \mathcal{C}^{3}(\mathbb {R})}$ is a double well symmetric potential, i.e. it satisfies W(?s) =  W(s) for ${s \in \mathbb {R},W(s) > 0}$ for ${s \in (-1,1)}$ , ${W(\pm 1) = 0}$ and ${W''(\pm 1) > 0}$ . Denoted with ${\theta_{2}}$ the saddle planar solution of (0.1), we show the existence of a unique solution ${\theta_{3} \in {\mathcal{C}^{2}}(\mathbb {R}^{3})}$ which is odd with respect to each variable, symmetric with respect to the diagonal planes, verifies ${0 < \theta_{3}(x,y,z) < 1}$ for x, y, z >  0 and ${\theta_{3}(x, y, z) \to_{z \to + \infty} \theta_{2}(x, y)}$ uniformly with respect to ${(x, y) \in \mathbb {R}^{2}}$ .     

Infinite-dimensional quasigroups of finite orders

Let Σ be a finite set of cardinality k > 0, let $\mathbb{A}$ be a finite or infinite set of indices, and let $\mathcal{F} \subseteq \Sigma ^\mathbb{A}$ be a subset consisting of finitely supported families. A function $f:\Sigma ^\mathbb{A} \to \Sigma$ is referred to as an $\mathbb{A}$ -quasigroup (if $\left| \mathbb{A} \right| = n$ , then an n-ary quasigroup) of order k if $f\left( {\bar y} \right) \ne f\left( {\bar z} \right)$ for any ordered families $\bar y$ and $\bar z$ that differ at exactly one position. It is proved that an $\mathbb{A}$ -quasigroup f of order 4 is reducible (representable as a superposition) or semilinear on every coset of $\mathcal{F}$ . It is shown that the quasigroups defined on Σ^?, where ? are positive integers, generate Lebesgue nonmeasurable subsets of the interval [0, 1].     

Algebras of Convolution Type Operators with Piecewise Slowly Oscillating Data. I: Local and Structural Study

Let ${\mathcal{B}_{p,w}}$ be the Banach algebra of all bounded linear operators acting on the weighted Lebesgue space ${L^{p}(\mathbb{R}, w)}$ , where ${p \in (1, \infty)}$ and w is a Muckenhoupt weight. We study the Banach subalgebra ${\mathfrak{A}_{p,w}}$ of ${\mathcal{B}_{p,w}}$ generated by all multiplication operators aI ( ${a \in PSO^{\diamond}}$ ) and all convolution operators W ⁰(b) ( ${b \in PSO_{p,w}^{\diamond}}$ ), where ${PSO^{\diamond} \subset L^{\infty}(\mathbb{R})}$ and ${PSO_{p,w}^{\diamond} \subset M_{p,w}}$ are algebras of piecewise slowly oscillating functions that admit piecewise slowly oscillating discontinuities at arbitrary points of ${\mathbb{R} \cup \{\infty\}}$ , and M _p,w is the Banach algebra of Fourier multipliers on ${L^{p}(\mathbb{R}, w)}$ . Under some conditions on the Muckenhoupt weight w, we construct a Fredholm symbol calculus for the Banach algebra ${\mathfrak{A}_{p,w}}$ and establish a Fredholm criterion for the operators ${A \in \mathfrak{A}_{p,w}}$ in terms of their Fredholm symbols. To study the Banach algebra ${\mathfrak{A}_{p,w}}$ we apply the theory of Mellin pseudodifferential operators, the Allan–Douglas local principle, the two idempotents theorem and the method of limit operators. The paper is divided in two parts. The first part deals with the local study of ${\mathfrak{A}_{p,w}}$ and necessary tools for studying local algebras.