设$A$是Hilbert空间（\h，<，>）中的正内射算子，用[，]表示由A：[f，g]=<Af，g>定义的内积.闭子空间$\s \subset \h$称为A相容的，如果存在$\s$的闭补，它关于内积[，]与$\s$正交。等价地，如果存在必然唯一的幂等算子$Q_\s$，使得$R(Q_\s)=\s$，它对于这个内积是对称的。相容格拉斯曼$Gr_A$是$\h$的所有A相容子空间的集合。通过参数化它通过一对一的对应$\s\leftrightarrow Q_\s$，这个集合被证明是一个可微子流形的Banach空间的所有运营商在$\h$是对称的形式[，]。一个Banach-李群自然地作用在相容格拉斯曼算子上，相容格拉斯曼算子是$\h$中所有可逆算子的群，它们保持形式[，]。有限维相容子空间$\s$的$Gr_A$中的每个连通分支都是Banach Lie-Poisson空间中的辛叶。对于$1\le p \le \infty$，在存在一个固定的[，]-正交分解的$\h$，$\h=\s_0\dot{+} \n_0$，我们研究了限制兼容格拉斯曼（一个类似的限制，或佐藤格拉斯曼）。本文证明了这一限制相容Grassmannian是p-Schatten算子Banach空间的一个子流形，该算子对于形式[，]是对称的。它带有可逆算子的Banach-李群的局部传递作用，该可逆算子保持[，]，并且具有G=1+K的形式，其中K在p-Schatten类中。利用投影对的Fredholm指数刻画了这类限制Grassmannian的连通分支。对于相容的Grassmannian，引入了与群作用等距的Finsler度量，证明了曲线的极小性结果.
英文摘要：
Let $A$ be a positive injective operator in a Hilbert space (\h, <,>), and denote by [,] the inner product defined by A: [f,g]=<Af,g>. A closed subspace $\s \subset \h$ is called A-compatible if there exists a closed complement for $\s$, which is orthogonal to $\s$ with respect to the inner product [,]. Equivalently, if there exists a necessarily unique idempotent operator $Q_\s$ such that $R(Q_\s)=\s$, which is symmetric for this inner product. The compatible Grassmannian $Gr_A$ is the set of all A-compatible subspaces of $\h$. By parametrizing it via the one to one correspondence $\s\leftrightarrow Q_\s$, this set is shown to be a differentiable submanifold of the Banach space of all operators in $\h$ which are symmetric with respect to the form [,]. A Banach-Lie group acts naturally on the compatible Grassmannian, the group of all invertible operators in $\h$ which preserve the form [,]. Each connected component in $Gr_A$ of a compatible subspace $\s$ of finite dimension, turns out to be a symplectic leaf in a Banach Lie-Poisson space. For $1\le p \le \infty$, in the presence of a fixed [,]-orthogonal decomposition of $\h$, $\h=\s_0\dot{+} \n_0$, we study the restricted compatible Grassmannian (an analogue of the restricted, or Sato Grassmannian). This restricted compatible Grassmannian is shown to be a submanifold of the Banach space of p-Schatten operators which are symmetric for the form [,]. It carries the locally transitive action of the Banach-Lie group of invertible operators which preserve [,], and are of the form G=1+K, with K in p-Schatten class. The connected components of this restricted Grassmannian are characterized by means of of the Fredholm index of pairs of projections. Finsler metrics which are isometric for the group actions are introduced for both compatible Grassmannians, and minimality results for curves are proved.
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Geometric characterization of Generalized Property (II)
Loyola University（洛约拉大学）；National Institute of Science Educational and Research Bhubaneswar, An OCC of Homi Bhabha National Institute（布巴内斯瓦尔国立科学教育研究中心，霍米·巴巴国立学院）
We construct a new bounded functional calculus for the generators of bounded semigroups on Hilbert spaces and generators of bounded holomorphic semigroups on Banach spaces. The calculus is a natural (and strict) extension of the classical Hille-Phillips functional calculus, and it is compatible with the other well-known functional calculi. It satisfies the standard properties of functional calculi, provides a unified and direct approach to a number of norm-estimates in the literature, and allows improvements of some of them.
Motivated by the definition of a semigroup compactification of a locally compact group and a large collection of examples, we introduce the notion of an (operator) "homogeneous left dual Banach algebra" (HLDBA) over a (completely contractive) Banach algebra $A$. We prove a Gelfand-type representation theorem showing that every HLDBA over $A$ has a concrete realization as an (operator) homogeneous left Arens product algebra: the dual of a subspace of $A^*$ with a compatible (matrix) norm and a type of left Arens product ${\scriptscriptstyle \square}$. Examples include all left Arens product alg…
In this work, we investigate measurement incompatibility in general probabilistic theories (GPTs). We show several equivalent characterizations of compatible measurements. The first is in terms of the positivity of associated maps. The second relates compatibility to the inclusion of certain generalized spectrahedra. For this, we extend the theory of free spectrahedra to ordered vector spaces. The third characterization connects the compatibility of dichotomic measurements to the ratio of tensor crossnorms of Banach spaces. We use these characterizations to study the amount of incompatibility …
We construct two bounded functional calculi for sectorial operators on Banach spaces, which enhance the functional calculus for analytic Besov functions, by extending the class of functions, generalizing and sharpening estimates, and adapting the calculus to the angle of sectoriality. The calculi are based on appropriate reproducing formulas, they are compatible with standard functional calculi and they admit appropriate convergence lemmas and spectral mapping theorems. To achieve this, we develop the theory of associated function spaces in ways which are interesting and significant. As conseq…
Duality for Interpolation Spaces Defined Via Slowly Varying Functions: The Case 0<q<1
S.G.T.B Khalsa College, University of Delhi（德里大学 S.G.T.B 卡尔萨学院）；Departamento de Matemáticas Facultad de Matemáticas Universidad de Murcia（穆尔西亚大学数学学院数学系）
We prove novel results on interpolation of Fredholm operators including an abstract factorization theorem. The main result of this paper provides sufficient conditions on the parameters $θ\in (0,1)$ and $q\in \lbrack 1,\infty ]$ under which an operator $A$ is a Fredholm operator from the real interpolation space $(X_{0},X_{1})_{θ,q}$ to $(Y_{0},Y_{1})_{θ,q} $ for a given operator $A\colon (X_{0},X_{1})\rightarrow (Y_{0},Y_{1})$ between compatible pairs of Banach spaces such that its restrictions to the endpoint spaces are Fredholm operators. These conditions are expressed in terms of the corre…
Two themes drive this article: identifying the structure necessary to formulate quaternionic operator theory and revealing the relation between complex and quaternionic operator theory. The theory of quaternionic right linear operators is usually formulated assuming the existenc of both a right- and a left-multiplication on the Banach space $V$, as the space of bounded operators on $V$ is otherwise not a quaternionic linear space. A right linear operator is however only associated with the right-multiplication and in certain settings, e.g. on Hilbert spaces, the left-multiplication is not defi…
