Let $(X,d)$ be a metric space and $J\subseteq [0,\infty)$ be
nonempty. We study the structure of the arbitrary intersections
of
Lipschitz algebras, and define a special Banach subalgebra of
$\bigcap_{\gamma\in J}\operatorname{Lip}_\gamma X$, denoted by
$\operatorname{ILip}_J X$. Mainly,
we investigate $C$-character amenability of $\operatorname{ILip}_J X$, in
particular Lipschitz algebras. We address a gap in the proof
of a
recent result in this field. Then we remove this gap, and obtain
a
necessary and sufficient condition for $C$-character amenability
of $\operatorname{ILip}_J X$, specially Lipschitz algebras, under an additional
assumption.
In this paper, we show that the Möbius invariant
function space $\mathcal {Q}_p$ can be generated by variant
Dirichlet type spaces
$\mathcal{D}_{\mu, p}$ induced by finite positive Borel measures
$\mu$ on the open unit disk. A criterion for the equality between
the space $\mathcal{D}_{\mu, p}$ and the usual Dirichlet type
space $\mathcal {D}_p$ is given. We obtain a sufficient condition
to construct different $\mathcal{D}_{\mu, p}$ spaces
and we provide examples.
We establish decomposition theorems for $\mathcal{D}_{\mu,
p}$ spaces, and prove that the non-Hilbert space $\mathcal
{Q}_p$ is equal to the intersection of Hilbert spaces $\mathcal{D}_{\mu,
p}$. As an application of the relation between $\mathcal {Q}_p$
and $\mathcal{D}_{\mu, p}$ spaces, we also obtain that there
exist different $\mathcal{D}_{\mu, p}$ spaces; this is a trick
to prove the existence without constructing examples.
We shall use the classical Perron envelope method to show a general
existence theorem to degenerate complex
Monge-Ampère type equations on compact Kähler manifolds.
Let $1\leq p\lt \infty$, and let $G$ be a discrete group. We give
a sufficient and necessary condition
for weighted translation operators on the Lebesgue space $\ell^p(G)$
to be densely disjoint hypercyclic.
The characterization for the dual of a weighted translation to
be densely disjoint hypercyclic is also obtained.
Let $R$ be a prime ring with extended
centroid $C$, $Q$ maximal right ring of quotients of $R$, $RC$
central closure of $R$ such that $dim_{C}(RC)
\gt 4$, $f(X_{1},\dots,X_{n})$
a multilinear polynomial over $C$ which is not central-valued
on $R$ and $f(R)$ the set of all evaluations of the multilinear
polynomial $f\big(X_{1},\dots,X_{n}\big)$ in $R$. Suppose that
$G$ is a nonzero generalized derivation of $R$ such that $G^2\big(u\big)u
\in C$ for all $u\in f(R)$ then one of the following conditions
holds:
(I) there exists $a\in Q$ such that $a^2=0$ and
either $G(x)=ax$ for all $x\in R$ or $G(x)=xa$ for all $x\in
R$;
(II) there exists $a\in Q$ such that $0\neq a^2\in
C$ and either $G(x)=ax$ for all $x\in R$ or $G(x)=xa$ for all
$x\in R$ and $f(X_{1},\dots,X_{n})^{2}$ is central-valued on
$R$;
(III) $char(R)=2$ and one of the following holds:
(i) there exist $a, b\in Q$ such that $G(x)=ax+xb$ for all
$x\in R$ and $a^{2}=b^{2}\in C$;
(ii) there exist $a, b\in Q$ such that $G(x)=ax+xb$ for all
$x\in R$, $a^{2}, b^{2}\in C$ and $f(X_{1},\ldots,X_{n})^{2}$
is central-valued on $R$;
(iii) there exist $a \in Q$ and an $X$-outer derivation $d$
of $R$ such that $G(x)=ax+d(x)$ for all $x\in R$, $d^2=0$ and
$a^2+d(a)=0$;
(iv) there exist $a \in Q$ and an $X$-outer derivation $d$
of $R$ such that $G(x)=ax+d(x)$ for all $x\in R$, $d^2=0$,
$a^2+d(a)\in C$ and $f(X_{1},\dots,X_{n})^{2}$ is central-valued
on $R$.
Moreover, we characterize the form of nonzero generalized derivations
$G$ of $R$ satisfying $G^2(x)=\lambda x$ for all $x\in R$, where
$\lambda \in C$.
Suppose $G$ is a connected complex Lie group and $H$ is a closed
complex subgroup.
Then there exists a closed complex subgroup $J$ of $G$ containing
$H$ such that
the fibration $\pi:G/H \to G/J$ is the holomorphic reduction
of $G/H$, i.e., $G/J$ is holomorphically
separable and ${\mathcal O}(G/H) \cong \pi^*{\mathcal O}(G/J)$.
In this paper we prove that if $G/H$ is pseudoconvex, i.e.,
if
$G/H$ admits a continuous plurisubharmonic exhaustion function,
then $G/J$ is Stein and $J/H$ has no non--constant holomorphic
functions.
We study linear projections on Plücker space whose restriction
to the Grassmannian is a non-trivial branched
cover.
When an automorphism of the Grassmannian preserves the fibers,
we show that the Grassmannian is necessarily
of $m$-dimensional linear subspaces in a symplectic vector
space of dimension $2m$, and the linear map is
the Lagrangian involution.
The Wronski map for a self-adjoint linear differential operator
and pole placement map for
symmetric linear systems are natural examples.
Fix an irreducible (finite) root system $R$ and a choice
of positive roots. For any algebraically closed field $k$ consider the almost simple, simply connected algebraic group $G_k$ over $k$ with root system $k$. One associates to any dominant weight $\lambda$ for $R$ two $G_k$--modules with highest weight $\lambda$, the
Weyl module $V (\lambda)_k$ and its simple quotient $L (\lambda)_k$.
Let $\lambda$ and $\mu$ be dominant weights with $\mu \lt \lambda$ such
that
$\mu$ is maximal with this property. Garibaldi, Guralnick, and
Nakano
have asked under which condition there exists $k$ such that $L
(\mu)_k$
is a composition factor of $V (\lambda)_k$, and they exhibit an
example
in type $E_8$ where this is not the case. The purpose of this
paper
is to to show that their example is the only one. It contains
two proofs
for this fact, one that uses a classification of the possible
pairs $(\lambda, \mu)$,
and another one that relies only on the classification
of root systems.
Alfred Schild has established conditions
that Lorentz transformations map world-vectors $(ct,x,y,z)$ with
integer coordinates onto vectors of the same kind. These transformations
are called integral Lorentz transformations.
The present paper contains supplements to
our earlier work
with a new focus on group theory. To relate the results to the
familiar matrix group nomenclature we associate Lorentz transformations
with matrices in $\mathrm{SL}(2,\mathbb{C})$. We consider the
lattice of subgroups of the group originated in Schild's paper
and obtain generating sets for the full group and its subgroups.
A $C^{*}$-algebra $A$ has the ideal property if any ideal
$I$ of $A$ is generated as a closed two sided ideal by the projections
inside the ideal. Suppose that the limit $C^{*}$-algebra $A$
of inductive limit of direct sums of matrix algebras over spaces
with uniformly bounded dimension has ideal property. In this
paper we will prove that $A$ can be written as an inductive limit
of certain very special subhomogeneous algebras, namely, direct
sum of dimension drop interval algebras and matrix algebras over
2-dimensional spaces with torsion $H^{2}$ groups.
It is known that every Toeplitz matrix $T$ enjoys a circulant
and skew circulant splitting (denoted by CSCS)
i.e., $T=C-S$ with $C$ a circulant matrix and $S$ a skew circulant
matrix. Based on the variant of such a splitting (also referred
to as CSCS), we first develop classical CSCS iterative methods
and then introduce shifted CSCS iterative methods for solving
hermitian positive definite Toeplitz systems in this paper. The
convergence of each method is analyzed. Numerical experiments
show that the classical CSCS iterative methods work slightly
better than the Gauss-Seidel (GS) iterative methods if the CSCS
is convergent, and that there is always a constant $\alpha$ such
that the shifted CSCS iteration converges much faster than the
Gauss-Seidel iteration, no matter whether the CSCS itself is
convergent or not.
In this paper, we obtain some characterizations of the (strong)
Birkhoff--James orthogonality for elements of Hilbert $C^*$-modules
and certain elements of $\mathbb{B}(\mathscr{H})$.
Moreover, we obtain a kind of Pythagorean relation for bounded
linear operators.
In addition, for $T\in \mathbb{B}(\mathscr{H})$ we prove that if the
norm attaining
set $\mathbb{M}_T$ is a unit sphere of some finite dimensional
subspace $\mathscr{H}_0$ of $\mathscr{H}$ and $\|T\|_{{{\mathscr{H}}_0}^\perp}
\lt \|T\|$, then for every $S\in\mathbb{B}(\mathscr{H})$, $T$ is the strong
Birkhoff--James orthogonal to $S$ if and only if there exists
a unit vector $\xi\in {\mathscr{H}}_0$ such that $\|T\|\xi =
|T|\xi$ and $S^*T\xi = 0$.
Finally, we introduce a new type of approximate orthogonality
and investigate this notion in the setting of inner product $C^*$-modules.
It is known that a bi-orderable group has no generalized torsion
element,
but the converse does not hold in general.
We conjecture that the converse holds for the fundamental groups
of $3$-manifolds,
and verify the conjecture for non-hyperbolic, geometric $3$-manifolds.
We also confirm the conjecture for some infinite families of
closed hyperbolic $3$-manifolds.
In the course of the proof,
we prove that each standard generator of the Fibonacci group
$F(2, m)$ ($m \gt 2$) is a generalized torsion element.
Let the measure algebra of a topological group $G$ be equipped
with
the topology of uniform convergence on bounded right uniformly
equicontinuous sets of functions.
Convolution is separately continuous on the measure algebra,
and it is jointly continuous if and only if $G$ has the SIN property.
On the larger space $\mathsf{LUC}(G)^\ast$ which includes the measure
algebra,
convolution is also jointly continuous if and only if the group
has the SIN property,
but not separately continuous for many non-SIN groups.
Let $R$
be a ring and $b, c\in R$.
In this paper, we give some characterizations of the $(b,c)$-inverse,
in terms of the direct sum decomposition, the annihilator and
the invertible elements.
Moreover, elements with equal $(b,c)$-idempotents related to
their $(b, c)$-inverses are characterized, and the reverse order
rule for the $(b,c)$-inverse is considered.
In this paper, we construct two classes of rational function
operators by using the Poisson integrals of the function on the
whole real
axis. The convergence rates of the uniform and mean approximation
of such rational function operators on the whole real axis are
studied.
For any ring $R$, we show that, in the bounded derived category
$D^{b}(\operatorname{Mod} R)$ of left $R$-modules,
the subcategory of complexes with finite Gorenstein projective
(resp. injective) dimension modulo the subcategory
of complexes with finite projective (resp. injective) dimension
is equivalent to
the stable category $\underline{\mathbf{GP}}(\operatorname{Mod} R)$ (resp.
$\overline{\mathbf{GI}}(\operatorname{Mod} R)$)
of Gorenstein projective (resp. injective) modules. As a consequence,
we get that if $R$ is a left and right noetherian ring admitting
a dualizing complex,
then $\underline{\mathbf{GP}}(\operatorname{Mod} R)$ and
$\overline{\mathbf{GI}}(\operatorname{Mod}
R)$ are equivalent.