Spherically Balanced Hilbert Spaces of Formal Power Series in Several Variables-II

We continue the study of spherically balanced Hilbert spaces initiated in the first part of this paper. Recall that the complex Hilbert space \(H^2(\beta )\) of formal power series in the variables \(z_1, \ldots , z_m\) is spherically balanced if and only if there exist a Reinhardt measure \(\mu \) supported on the unit sphere \(\partial {\mathbb {B}}\) and a Hilbert space \(H^2(\gamma )\) of formal power series in the variable \(t\) such that

$$\begin{aligned} \Vert f\Vert ^2_{H^2(\beta )} = \int _{\partial {\mathbb {B}}}\Vert {f_z}\Vert ^2_{H^2(\gamma )}~d\mu (z)~(f \in H^2(\beta )), \end{aligned}$$

where \(f_z(t)=f(t z)\) is a formal power series in the variable \(t\). In the first half of this paper, we discuss operator theory in spherically balanced Hilbert spaces. The first main result in this part describes quasi-similarity orbit of multiplication tuple \(M_z\) on a spherically balanced space \(H^2(\beta ).\) We also observe that all spherical contractive multi-shifts on spherically balanced spaces admit the classical von Neumann’s inequality. In the second half, we introduce and study a class of Hilbert spaces, to be referred to as \({\mathcal {G}}\)-balanced Hilbert spaces, where \({\mathcal {G}}={\mathcal {U}}(r_1) \times {\mathcal {U}}(r_2) \times \cdots \times {\mathcal {U}}(r_k)\) is a subgroup of \({\mathcal {U}}(m)\) with \(r_1 + \cdots + r_k=m.\) In the case in which \({\mathcal {G}}={\mathcal {U}}(m),\) \({\mathcal {G}}\)-balanced spaces are precisely spherically balanced Hilbert spaces.

     

Tempered representations of <Emphasis Type="Italic">p</Emphasis>-adic groups: special idempotents and topology

Let \(G=G(k)\) be a connected reductive group over a p-adic field k. The smooth (and tempered) complex representations of G can be considered as the nondegenerate modules over the Hecke algebra \({\mathcal {H}}={\mathcal {H}}(G)\) and the Schwartz algebra \({\mathcal {S}}={\mathcal {S}}(G)\) forming abelian categories \({\mathcal {M}}(G)\) and \({\mathcal {M}}^t(G)\), respectively. Idempotents \(e\in {\mathcal {H}}\) or \({\mathcal {S}}\) define full subcategories \({\mathcal {M}}_e(G)= \{V : {\mathcal {H}}eV=V\}\) and \({\mathcal {M}}_e^t(G)= \{V : {\mathcal {S}}eV=V\}\). Such an e is said to be special (in \({\mathcal {H}}\) or \({\mathcal {S}}\)) if the corresponding subcategory is abelian. Parallel to Bernstein’s result for \(e\in {\mathcal {H}}\) we will prove that, for special \(e \in {\mathcal {S}}\), \({\mathcal {M}}_e^t(G) = \prod _{\Theta \in \theta _e} {\mathcal {M}}^t(\Theta )\) is a finite direct product of component categories \({\mathcal {M}}^t(\Theta )\), now referring to connected components of the center of \({\mathcal {S}}\). A special \(e\in {\mathcal {H}}\) will be also special in \({\mathcal {S}}\), but idempotents \(e\in {\mathcal {H}}\) not being special can become special in \({\mathcal {S}}\). To obtain conditions we consider the sets \(\mathrm{Irr}^t(G) \subset \mathrm{Irr}(G)\) of (tempered) smooth irreducible representations of G, and we view \(\mathrm{Irr}(G)\) as a topological space for the Jacobson topology defined by the algebra \({\mathcal {H}}\). We use this topology to introduce a preorder on the connected components of \(\mathrm{Irr}^t(G)\). Then we prove that, for an idempotent \(e \in {\mathcal {H}}\) which becomes special in \({\mathcal {S}}\), its support \(\theta _e\) must be saturated with respect to that preorder. We further analyze the above decomposition of \({\mathcal {M}}_e^t(G)\) in the case where G is k-split with connected center and where \(e = e_J \in {\mathcal {H}}\) is the Iwahori idempotent. Here we can use work of Kazhdan and Lusztig to relate our preorder on the support \(\theta _{e_J}\) to the reverse of the natural partial order on the unipotent classes in G. We finish by explicitly computing the case \(G=GL_n\), where \(\theta _{e_J}\) identifies with the set of partitions of n. Surprisingly our preorder (which is a partial order now) is strictly coarser than the reverse of the dominance order on partitions.     

Walk-Powers and Homomorphism Bounds of Planar Signed Graphs

As an extension of the Four-Color Theorem it is conjectured by the first author that every planar graph of odd-girth at least \(2k+1\) admits a homomorphism to the projective cube of dimension 2k, i.e., the Cayley graph \({\mathcal {PC}}(2k)=({\mathbb {Z}}_2^{2k}, \{e_1, e_2,\) \(\ldots ,e_{2k}, J\})\) where the \(e_i\)’s are the standard basis vectors of \({\mathbb {Z}}_2^d\) and J is the all 1 vector. Noting that \({\mathcal {PC}}(2k)\) itself is of odd-girth \(2k+1\), in this work we show that if the conjecture is true, then \({\mathcal {PC}}(2k)\) is an optimal such graph both with respect to the number of vertices and the number of edges. The result is obtained using the notion of walk-power of graphs and their clique numbers. An analogous result is proved for signed bipartite planar graphs of unbalanced-girth 2k. The work is presented in the uniform framework of planar consistent signed graphs.     

Derivations,Local and 2-Local Derivations on Some Algebras of Operators on Hilbert C*-Modules

For a commutative C*-algebra \({\mathcal {A}}\) with unit e and a Hilbert \({\mathcal {A}}\)-module \({\mathcal {M}}\), denote by End\(_{{\mathcal {A}}}({\mathcal {M}})\) the algebra of all bounded \({\mathcal {A}}\)-linear mappings on \({\mathcal {M}}\), and by End\(^*_{{\mathcal {A}}}({\mathcal {M}})\) the algebra of all adjointable mappings on \({\mathcal {M}}\). We prove that if \({\mathcal {M}}\) is full, then each derivation on End\(_{{\mathcal {A}}}({\mathcal {M}})\) is \({\mathcal {A}}\)-linear, continuous, and inner, and each 2-local derivation on End\(_{{\mathcal {A}}}({\mathcal {M}})\) or End\(^{*}_{{\mathcal {A}}}({\mathcal {M}})\) is a derivation. If there exist \(x_0\) in \({\mathcal {M}}\) and \(f_0\) in \({\mathcal {M}}^{'}\), such that \(f_0(x_0)=e\), where \({\mathcal {M}}^{'}\) denotes the set of all bounded \({\mathcal {A}}\)-linear mappings from \({\mathcal {M}}\) to \({\mathcal {A}}\), then each \({\mathcal {A}}\)-linear local derivation on End\(_{{\mathcal {A}}}({\mathcal {M}})\) is a derivation.     

Structure theorem of the generator of a norm continuous completely positive semigroup: an alternative proof using Bures distance

Here we present an alternative proof using Bures distance that the generator L of a norm continuous completely positive semigroup acting on a \(C^*\)-algebra \({\mathcal {B}}\subset \mathcal B(H)\) has the form \( L(b) = \Psi (b) + k^*b+bk\), \(b\in {\mathcal {B}}\) for some completely positive map \(\Psi :{\mathcal {B}}\rightarrow {\mathcal {B}}(H)\) and \(k\in {\mathcal {B}}(H)\).     

Semigroups of rectangular matrices under a sandwich operation

Let \({\mathcal {M}}_{mn}={\mathcal {M}}_{mn}({\mathbb {F}})\) denote the set of all \(m\times n\) matrices over a field \({\mathbb {F}}\), and fix some \(n\times m\) matrix \(A\in {\mathcal {M}}_{nm}\). An associative operation \(\star \) may be defined on \({\mathcal {M}}_{mn}\) by \(X\star Y=XAY\) for all \(X,Y\in {\mathcal {M}}_{mn}\), and the resulting sandwich semigroup is denoted \({\mathcal {M}}_{mn}^A={\mathcal {M}}_{mn}^A({\mathbb {F}})\). These semigroups are closely related to Munn rings, which are fundamental tools in the representation theory of finite semigroups. We study \({\mathcal {M}}_{mn}^A\) as well as its subsemigroups \(\hbox {Reg}({\mathcal {M}}_{mn}^A)\) and \({\mathcal {E}}_{mn}^A\) (consisting of all regular elements and products of idempotents, respectively), and the ideals of \(\hbox {Reg}({\mathcal {M}}_{mn}^A)\). Among other results, we characterise the regular elements; determine Green’s relations and preorders; calculate the minimal number of matrices (or idempotent matrices, if applicable) required to generate each semigroup we consider; and classify the isomorphisms between finite sandwich semigroups \({\mathcal {M}}_{mn}^A({\mathbb {F}}_1)\) and \({\mathcal {M}}_{kl}^B({\mathbb {F}}_2)\). Along the way, we develop a general theory of sandwich semigroups in a suitably defined class of partial semigroups related to Ehresmann-style “arrows only” categories; we hope this framework will be useful in studies of sandwich semigroups in other categories. We note that all our results have applications to the variants \({\mathcal {M}}_n^A\) of the full linear monoid \({\mathcal {M}}_n\) (in the case \(m=n\)), and to certain semigroups of linear transformations of restricted range or kernel (in the case that \(\hbox {rank}(A)\) is equal to one of m, n).     

An Invariant Subspace Theorem and Invariant Subspaces of Analytic Reproducing Kernel Hilbert Spaces - II

This paper is a follow-up contribution to our work (Sarkar in J Oper Theory, 73:433–441, 2015) where we discussed some invariant subspace results for contractions on Hilbert spaces. Here we extend the results of (Sarkar in J Oper Theory, 73:433–441, 2015) to the context of n-tuples of bounded linear operators on Hilbert spaces. Let \(T = (T_1, \ldots , T_n)\) be a pure commuting co-spherically contractive n-tuple of operators on a Hilbert space \({\mathcal {H}}\) and \({\mathcal {S}}\) be a non-trivial closed subspace of \({\mathcal {H}}\). One of our main results states that: \({\mathcal {S}}\) is a joint T-invariant subspace if and only if there exists a partially isometric operator \(\Pi \in {\mathcal {B}}(H^2_n({\mathcal {E}}), {\mathcal {H}})\) such that \({\mathcal {S}}= \Pi H^2_n({\mathcal {E}})\), where \(H^2_n\) is the Drury–Arveson space and \({\mathcal {E}}\) is a coefficient Hilbert space and \(T_i \Pi = \Pi M_{z_i}\), \(i = 1, \ldots , n\). In particular, it follows that a shift invariant subspace of a “nice” reproducing kernel Hilbert space over the unit ball in \({{\mathbb {C}}}^n\) is the range of a “multiplier” with closed range. Our work addresses the case of joint shift invariant subspaces of the Hardy space and the weighted Bergman spaces over the unit ball in \({{\mathbb {C}}}^n\).     

Finite elements in some vector lattices of nonlinear operators

We study the collection of finite elements \(\Phi _{1}\big ({\mathcal {U}}(E,F)\big )\) in the vector lattice \({\mathcal {U}}(E,F)\) of orthogonally additive, order bounded (called abstract Uryson) operators between two vector lattices E and F, where F is Dedekind complete. In particular, for an atomic vector lattice E it is proved that for a finite element in \(\varphi \in {\mathcal {U}}(E,{\mathbb {R}})\) there is only a finite set of mutually disjoint atoms, where \(\varphi \) does not vanish and, for an atomless vector lattice the zero-vector is the only finite element in the band of \(\sigma \)-laterally continuous abstract Uryson functionals. We also describe the ideal \(\Phi _{1}\big ({\mathcal {U}}({\mathbb {R}}^n,{\mathbb {R}}^m)\big )\) for \(n,m\in {\mathbb {N}}\) and consider rank one operators to be finite elements in \({\mathcal {U}}(E,F)\).     

Linear extensions of some Baire-one functions

Let X be a Hausdorff topological space, and let \({\mathscr {B}}_1(X)\) denote the space of all real Baire-one functions defined on X. Let A be a nonempty subset of X endowed with the topology induced from X, and let \({\mathscr {F}}(A)\) be the set of functions \(A\rightarrow {\mathbb R}\) with a property \({\mathscr {F}}\) making \({\mathscr {F}}(A)\) a linear subspace of \({\mathscr {B}}_1(A)\). We give a sufficient condition for the existence of a linear extension operator \(T_A:{\mathscr {F}}(A)\rightarrow {\mathscr {F}}(X)\), where \({\mathscr {F}}\) means to be piecewise continuous on a sequence of closed and \(G_\delta \) subsets of X and is denoted by \({\mathscr {P}_0}\). We show that \(T_A\) restricted to bounded elements of \({\mathscr {F}}(A)\) endowed with the supremum norm is an isometry. As a consequence of our main theorem, we formulate the conclusion about existence of a linear extension operator for the classes of Baire-one-star and piecewise continuous functions.     

Vizing Bound for the Chromatic Number on Some Graph Classes

We are interested in hereditary classes of graphs \({\mathcal {G}}\) such that every graph \(G \in {\mathcal {G}}\) satisfies \(\varvec{\chi }(G) \le \omega (G) + 1\), where \(\chi (G)\) (\(\omega (G)\)) denote the chromatic (clique) number of G. This upper bound is called the Vizing bound for the chromatic number. Apart from perfect graphs few classes are known to satisfy the Vizing bound in the literature. We show that if G is (\(P_6, S_{1, 2, 2}\), diamond)-free, then \(\chi (G) \le \omega (G)+1\), and we give examples to show that the bound is sharp.     

On Summation of Nonharmonic Fourier Series

Let a sequence \(\Lambda \subset {\mathbb {C}}\) be such that the corresponding system of exponential functions \({\mathcal {E}}(\Lambda ):=\left\{ {\text {e}}^{i\lambda t}\right\} _{\lambda \in \Lambda }\) is complete and minimal in \(L^2(-\pi ,\pi )\), and thus each function \(f\in L^2(-\pi ,\pi )\) corresponds to a nonharmonic Fourier series in \({\mathcal {E}}(\Lambda )\). We prove that if the generating function \(G\) of \(\Lambda \) satisfies the Muckenhoupt \((A_2)\) condition on \({\mathbb {R}}\), then this series admits a linear summation method. Recent results show that the \((A_2)\) condition cannot be omitted.     

Extremal Problems for Mappings with Generalized Parametric Representation in $${{\mathbb {C}}}^{n}$$

In this paper we are concerned with the family \(\widetilde{S}^t_A(\mathbb {B}^n)\) (\(t\ge 0\)) of normalized biholomorphic mappings on the Euclidean unit ball \(\mathbb {B}^n\) in \({\mathbb {C}}^n\) that can be embedded in normal Loewner chains whose normalizations are given by time-dependent operators \(A\in \widetilde{\mathcal {A}}\), where \(\widetilde{\mathcal {A}}\) is a family of measurable mappings from \([0,\infty )\) into \(L({\mathbb {C}}^n)\) which satisfy certain natural assumptions. In particular, we consider extreme points and support points associated with the compact family \(\widetilde{S}^t_A(\mathbb {B}^n)\), where \(A\in \widetilde{\mathcal {A}}\). We prove that if \(f(z,t)=V(t)^{-1}z+\cdots \) is a normal Loewner chain such that \(V(s)f(\cdot ,s)\in \mathrm{ex}\,\widetilde{S}^s_A(\mathbb {B}^n)\) (resp. \(V(s)f(\cdot ,s)\in \mathrm{supp}\,\widetilde{S}^s_A(\mathbb {B}^n)\)), then \(V(t)f(\cdot ,t)\in \mathrm{ex}\, \widetilde{S}^t_A(\mathbb {B}^n)\), for all \(t\ge s\) (resp. \(V(t)f(\cdot ,t)\in \mathrm{supp}\,\widetilde{S}^t_A(\mathbb {B}^n)\), for all \(t\ge s\)), where V(t) is the unique solution on \([0,\infty )\) of the initial value problem: \(\frac{d V}{d t}(t)=-A(t)V(t)\), a.e. \(t\ge 0\), \(V(0)=I_n\). Also, we obtain an example of a bounded support point for the family \(\widetilde{S}_A^t(\mathbb {B}^2)\), where \(A\in \widetilde{\mathcal {A}}\) is a certain time-dependent operator. We also consider the notion of a reachable family with respect to time-dependent linear operators \(A\in \widetilde{\mathcal {A}}\), and obtain characterizations of extreme/support points associated with these families of bounded biholomorphic mappings on \(\mathbb {B}^n\). Useful examples and applications yield that the study of the family \(\widetilde{S}^t_A(\mathbb {B}^n)\) for time-dependent operators \(A\in \widetilde{\mathcal {A}}\) is basically different from that in the case of constant time-dependent linear operators.     

Homomorphisms of $$\ell ^1$$-Munn algebras and Rees matrix semigroup algebras

Let \({\mathcal {LM}}\left( {\mathcal {A}}, P\right) \) be an \(\ell ^1\)-Munn algebra over an arbitrary unital Banach algebra \({\mathcal {A}}\). We characterize homomorphisms from \({\mathcal {LM}}\left( {\mathcal {A}}, P\right) \) into an arbitrary Banach algebra \({\mathcal {B}}\) in terms of homomorphisms from \({\mathcal {A}}\) into \({\mathcal {B}}\). Then we discuss homomorphisms from arbitrary Banach algebras into \({\mathcal {LM}}\left( {\mathcal {A}}, P\right) \). Existence and uniqueness of homomorphisms under certain conditions are also discussed. We apply these results to the concrete case of \(\ell ^1(S)\) where S is a Rees matrix semigroup, to identify characters of \(\ell ^1(S)\) in both cases where S is with or without zero. As a consequence if the sandwich matrix of S has a zero entry, then \(\ell ^1(S)\) is character amenable.     

Augmentation quotients for real representation rings of cyclic groups

Denote by \(C_m\) the cyclic group of order m. Let \({\mathcal {R}}(C_m)\) be its real representation ring, and \(\Delta (C_m)\) its augmentation ideal. In this paper, we give an explicit \({\mathbb {Z}}\)-basis for the n-th power \(\Delta ^{n}(C_m)\) and determine the isomorphism class of the n-th augmentation quotient \(\Delta ^n(C_m)/\Delta ^{n+1}(C_m)\) for each positive integer n.     

Families of optimal packings in real and complex Grassmannian spaces

A construction based on a \(4l \times 4l\) Hadamard matrix leads to a new family of optimal orthoplex packings in Grassmannian spaces \(G_{\mathbb {R}}(8l, 4l)\) and \(G_{\mathbb {C}}(4l, 2l)\). A related construction gives an optimal simplex packings in \(G_{\mathbb {R}}(8 l-1, 4 l - 1)\) and \(G_{\mathbb {R}}(8l-1, 4l)\) with the additional assumption of an \(8l \times 8l\) skew Hadamard matrix and a related 1-factorization of a complete graph. A construction of a maximal optimal simplex packings in \(G_{\mathbb {C}}(2l-1, l- 1)\) and \(G_{\mathbb {C}}(2l-1,l)\) is given.     

A normal generating set for the Torelli group of a compact non-orientable surface

For a compact surface S, let \({\mathcal {I}}(S)\) denote the Torelli group of S. For a compact orientable surface \(\Sigma \), \({\mathcal {I}}(\Sigma )\) is generated by two types of mapping classes, called bounding simple closed curve maps (BSCC maps) and bounding pair maps (BP maps) (see Powell in Proc Am Math Soc 68:347–350, 1978; Putman in Geom Topol 11:829–865, 2007). For a non-orientable closed surface N, \({\mathcal {I}}(N)\) is generated by BSCC maps and BP maps (see Hirose and Kobayashi in Fund Math 238:29–51, 2017). In this paper, we give an explicit normal generating set for \({\mathcal {I}}(N_g^b)\), where \(N_g^b\) is a genus-g compact non-orientable surface with b boundary components for \(g\ge 4\) and \(b\ge 1\).     

A geometric approach to alternating <Emphasis Type="Italic">k</Emphasis>-linear forms

Denote by \({{\mathcal {G}}}_k(V)\) the Grassmannian of the k-subspaces of a vector space V over a field \({\mathbb {K}}\). There is a natural correspondence between hyperplanes H of \({\mathcal {G}}_k(V)\) and alternating k-linear forms on V defined up to a scalar multiple. Given a hyperplane H of \({{\mathcal {G}}_k}(V)\), we define a subspace \(R^{\uparrow }(H)\) of \({{\mathcal {G}}_{k-1}}(V)\) whose elements are the \((k-1)\)-subspaces A such that all k-spaces containing A belong to H. When \(n-k\) is even, \(R^{\uparrow }(H)\) might be empty; when \(n-k\) is odd, each element of \({\mathcal {G}}_{k-2}(V)\) is contained in at least one element of \(R^{\uparrow }(H)\). In the present paper, we investigate several properties of \(R^{\uparrow }(H)\), settle some open problems and propose a conjecture.     

On decomposability of Multilinear sets

We consider the Multilinear set \({\mathcal {S}}\) defined as the set of binary points (x, y) satisfying a collection of multilinear equations of the form \(y_I = \prod _{i \in I} x_i\), \(I \in {\mathcal {I}}\), where \({\mathcal {I}}\) denotes a family of subsets of \(\{1,\ldots , n\}\) of cardinality at least two. Such sets appear in factorable reformulations of many types of nonconvex optimization problems, including binary polynomial optimization. A great simplification in studying the facial structure of the convex hull of the Multilinear set is possible when \({\mathcal {S}}\) is decomposable into simpler Multilinear sets \({\mathcal {S}}_j\), \(j \in J\); namely, the convex hull of \({\mathcal {S}}\) can be obtained by convexifying each \({\mathcal {S}}_j\), separately. In this paper, we study the decomposability properties of Multilinear sets. Utilizing an equivalent hypergraph representation for Multilinear sets, we derive necessary and sufficient conditions under which \({\mathcal {S}}\) is decomposable into \({\mathcal {S}}_j\), \(j \in J\), based on the structure of pair-wise intersection hypergraphs. Our characterizations unify and extend the existing decomposability results for the Boolean quadric polytope. Finally, we propose a polynomial-time algorithm to optimally decompose a Multilinear set into simpler subsets. Our proposed algorithm can be easily incorporated in branch-and-cut based global solvers as a preprocessing step for cut generation.     

Examples of Ricci-Mean Curvature Flows

Let \(\pi :{\mathbb {P}}({\mathcal {O}}(0)\oplus {\mathcal {O}}(k))\rightarrow {\mathbb {P}}^{n-1}\) be a projective bundle over \({\mathbb {P}}^{n-1}\) with \(1\le k \le n-1\). We denote \({\mathbb {P}}({\mathcal {O}}(0)\oplus {\mathcal {O}}(k))\) by \(N_{k}^{n}\) and endow it with the U(n)-invariant gradient shrinking Kähler Ricci soliton structure constructed by Cao (Elliptic and parabolic methods in geometry (Minneapolis, MN, 1994), A K Peters, Wellesley, 1996) and Koiso (Recent topics in differential and analytic geometry. Advanced studies in pure mathematics, Boston, 1990). In this paper, we show that lens space \(L(k\, ;1)(r)\) with radius r embedded in \(N_{k}^{n}\) is a self-similar solution. We also prove that there exists a pair of critical radii \(r_{1}<r_{2}\), which satisfies the following. The lens space \(L(k\, ;1)(r)\) is a self-shrinker if \(r<r_{2}\) and self-expander if \(r_{2}<r\), and the Ricci-mean curvature flow emanating from \(L(k\, ;1)(r)\) collapses to the 0-section of \(\pi \) if \(r<r_{1}\) and to the \(\infty \)-section of \(\pi \) if \(r_{1}<r\). This paper gives explicit examples of Ricci-mean curvature flows.