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We provide a corrected proof of one of the two main results (Theorem 3.2) of the paper "The space of persistence diagrams on n points coarsely embeds into Hilbert space" [Proc. Amer. Math. Soc. 149 (2021), pp. 2693–2703]. [ABSTRACT FROM AUTHOR]

Let H^n(a,b) be the one-dimensional Sobolev Hilbert space of order n on the interval (a,b) (n=1,2,\dots). The explicit form of the reproducing kernel of H^n(a,b) on finite interval was already known in the case for n=1,2,3. In this paper we obtain the explicit form of the reproducing kernel of H^n(a,b) on any interval for general n. [ABSTRACT FROM AUTHOR]

Let \mathcal {B}(H) be the algebra of all bounded operators on a Hilbert space H. Let T=V|T| be the polar decomposition of an operator T\in \mathcal {B}(H). The mean transform of T is defined by M(T)=\frac {T+|T|V}{2}. In this paper, we discuss several properties related to the spectrum, the kernel, the image, and the polar decomposition of mean transform. Moreover, we investigate the image and preimage by the mean transform of some class of operators such as positive, normal, unitary, hyponormal, and co-hyponormal operators. [ABSTRACT FROM AUTHOR]

For any real \beta let H^2_\beta be the Hardy-Sobolev space on the unit ball \mathbb {B}_{n}, n\geq 1. H^2_\beta is a reproducing kernel Hilbert space and its reproducing kernel is bounded when \beta >n/2. In this paper, we characterize when the composition operator C_{\varphi } on H^{2}_{\beta } is Fredholm for a non-constant analytic map \varphi :\mathbb {B}_{n}\to \mathbb {B}_{n}. [ABSTRACT FROM AUTHOR]

Let \mathcal {H} be a separable Hilbert space and let B(\mathcal {H}) be the *-algebra of all bounded linear operators on \mathcal {H}. In the present paper, we prove that a positive/regular operator from L_1(0,1) into an arbitrary separable operator ideal in B(\mathcal {H}) is necessarily Dunford–Pettis, extending and strengthening results due to Gretsky and Ostroy [Glasgow Math. J. 28 (1986), pp. 113–114], and Holub [Proc. Amer. Math. Soc. 104 (1988), pp. 89–95]. Consequently, for an arbitrary atomless von Neumann algebra \mathcal {M} and an arbitrary KB -ideal C_E in B(\mathcal {H}), the predual \mathcal {M}_* of \mathcal {M} is not isomorphic to any subspace of C_E. This observation complements several earlier results. [ABSTRACT FROM AUTHOR]

Recently, Simons provided a lemma for a support function of a closed convex set in a general Hilbert space and used it to prove the geometry conjecture on cycles of projections. In this paper, we extend Simons's lemma to closed convex functions, show its connections to Attouch–Théra duality, and use it to characterize (classical and phantom) cycles and gap vectors of proximal mappings. [ABSTRACT FROM AUTHOR]

In this paper, we give three different new proofs of the validity of the geometry conjecture about cycles of projections onto nonempty closed, convex subsets of a Hilbert space. The first uses a simple minimax theorem, which depends on the finite dimensional Hahn-Banach theorem. The second uses Fan's inequality, which has found many applications in optimization and mathematical economics. The third uses three results on maximally monotone operators on a Hilbert space. [ABSTRACT FROM AUTHOR]

A remarkable result of Molnár [Proc. Amer. Math. Soc. 126 (1998), pp. 853–861] states that automorphisms of the algebra of operators acting on a separable Hilbert space are stable under "small" perturbations. More precisely, if \phi,\psi are endomorphisms of \mathcal {B}(\mathcal {H}) such that \|\phi (A)-\psi (A)\|<\|A\| and \psi is surjective, then so is \phi. The aim of this paper is to extend this result to a larger class of Banach spaces including \ell _p and L_p spaces, where 1

We show that well-established methods from the theory of Banach modules and time-frequency analysis allow to derive completeness results for the collection of shifted and dilated version of a given (test) function in a quite general setting. While the basic ideas show strong similarity to the arguments used in a recent paper by V. Katsnelson we extend his results in several directions, both relaxing the assumptions and widening the range of applications. There is no need for the Banach spaces considered to be embedded into (L2(R), ||⋅||2), nor is the Hilbert space structure relevant. We choose to present the results in the setting of the Euclidean spaces, because then the Schwartz space S'(Rd) (d ≥ 1) of tempered distributions provides a well-established environment for mathematical analysis. We also establish connections to modulation spaces and Shubin classes (Qs(Rd), ||⋅||Qs), showing that they are special cases of Katsnelson's setting (only) for s ≥ 0. [ABSTRACT FROM AUTHOR]

By a result of Schur [J. Reine Angew. Math. 140 (1911), pp. 1–28], the entrywise product M ∘ N of two positive semidefinite matrices M,N is again positive. Vybíral [Adv. Math. 368 (2020), p. 9] improved on this by showing the uniform lower bound M ∘ M ≥ En/n for all n × n real or complex correlation matrices M, where En is the all-ones matrix. This was applied to settle a conjecture of Novak [J. Complexity 15 (1999), pp. 299–316] and to positive definite functions on groups. Vybíral (in his original preprint) asked if one can obtain similar uniform lower bounds for higher entrywise powers of M, or for M ∘ N when N ≠ M, M. A natural third question is to ask for a tighter lower bound that does not vanish as n → ∞, i.e., over infinite-dimensional Hilbert spaces. In this note, we affirmatively answer all three questions by extending and refining Vybíral's result to lower-bound M ∘ N, for arbitrary complex positive semidefinite matrices M,N. Specifically: we provide tight lower bounds, improving on Vybíral's bounds. Second, our proof is 'conceptual' (and self-contained), providing a natural interpretation of these improved bounds via tracial Cauchy–Schwarz inequalities. Third, we extend our tight lower bounds to Hilbert–Schmidt operators. As an application, we settle Open Problem 1 of Hinrichs–Krieg–Novak–Vybíral [J. Complexity 65 (2021), Paper No. 101544, 20 pp.], which yields improvements in the error bounds in certain tensor product (integration) problems. [ABSTRACT FROM AUTHOR]

In this paper, Lie group representations on Hilbert spaces are studied in relation with operator ranges. Let R be an operator range of a Hilbert space H. Given the set Λ of R-invariant operators, and given a Lie group representation ρ : G → GL(H), we discuss the induced semigroup homomorphism ρ : ρ−1(Λ) → B(R) for the operator range topology on R. In one direction, we work under the assumption ρ−1 (Λ) = G, so ρ : G →B(R) is in fact a group representation. In this setting, we prove that ρ is continuous (and smooth) if and only if the tangent map dρ is R-invariant. In another direction, we prove that for the tautological representations of unitary or invertible operators of an arbitrary infinite-dimensional Hilbert space H, the set ρ−1(Λ) is neither a group for a large set of nonclosed operator ranges R nor closed for all nonclosed operator ranges R. Both results are proved by means of explicit counterexamples. [ABSTRACT FROM AUTHOR]

We provide an elementary proof of a result by V. P. Fonf and C. Zanco on point-finite coverings of separable Hilbert spaces. Indeed, by using a variation of the famous argument introduced by J. Lindenstrauss and R. R. Phelps [Israel J. Math. 6 (1968), pp. 39–48] to prove that the unit ball of a reflexive infinite-dimensional Banach space has uncountably many extreme points, we prove the following result. Let X be an infinite-dimensional Hilbert space satisfying dens(X)< 2ℵ0, then X does not admit point-finite coverings by open or closed balls, each of positive radius. In the second part of the paper, we follow the argument introduced by V. P. Fonf, M. Levin, and C. Zanco in [J. Geom. Anal. 24 (2014), pp. 1891–1897] to prove that the previous result holds also in infinite-dimensional Banach spaces that are both uniformly rotund and uniformly smooth. [ABSTRACT FROM AUTHOR]

In this paper, we study a nonlinear time-fractional Volterra equation with nonsingular Mittag-Leffler kernel in Hilbert spaces. By applying the properties of Mittag-Leffler functions and the method of eigenvalue expansion, we give a mild solution of our problem. Our main tool here is using some Sobolev embeddings. [ABSTRACT FROM AUTHOR]

The unitary extension principle by Ron and Shen is one of the cornerstones of wavelet frame theory; it leads to tight frames for L2(R) and associated expansions of functions ƒ ∈ L2(R) of similar type as those for orthonormal wavelet bases. In this paper, the unitary extension principle is extended to the setting of a locally compact abelian group, equipped with a collection of nested co-compact subgroups. Unlike all previously known generalizations of the unitary extension principle, the current one is taking place within the setting of continuous frames, which means that the resulting decompositions of functions in the underlying Hilbert space in general are given in terms of integral representations rather than discrete sums. The frame elements themselves appear by letting a collection of modulation operators act on a countable family of basic functions. [ABSTRACT FROM AUTHOR]

Let π : G → B(E) be a continuous representation of a compact group G on a Banach space E. We prove that the set of vectors π (h)x, as h runs through the set T(G) of all trigonometric polynomials on G, and x runs through E, spans an invariant dense linear subspace of E. We prove the existence of a topological direct sum decomposition E = ⊕θ ∈ GEθ for E, where each Eθ is a closed π-invariant subspace of E. If λp : M(G) → B(Lp(G)), p ∈(1,∞), is the left regular representation of the measure algebra M(G) and B ⊂ PMp(G) is a homogeneous Banach space, we show that B λp(T(G)) is norm dense in B. Since Hilbert space techniques are not available, new machinery is developed in the paper for the proofs. [ABSTRACT FROM AUTHOR]

The wandering subspace problem for an analytic norm-increasing m-isometry T on a Hilbert space H asks whether every T-invariant subspace of H can be generated by a wandering subspace. An affirmative solution to this problem for m = 1 is ascribed to Beurling-Lax-Halmos, while that for m = 2 is due to Richter. In this paper, we capitalize on the idea of weighted shift on a one-circuit directed graph to construct a family of analytic cyclic 3-isometries which do not admit the wandering subspace property and which are norm-increasing on the orthogonal complement of a one-dimensional space. Further, on this one-dimensional space, their norms can be made arbitrarily close to 1. We also show that if the wandering subspace property fails for an analytic norm-increasing m-isometry, then it fails miserably in the sense that the smallest T-invariant subspace generated by the wandering subspace is of infinite codimension. [ABSTRACT FROM AUTHOR]

Let H be a complex, separable Hilbert space and let B(H) denote the algebra of all bounded linear operators acting on H. Given a unitarily-invariant norm |⋅|u on B(H) and two linear operators A and B in B(H), we shall say that A and B are polynomially isometric relative to |⋅|u if |p(A)|u = |p(B)|u for all polynomials p. In this paper, we examine to what extent an operator A being polynomially isometric to a normal operator N implies that A is itself normal. More explicitly, we first show that if |⋅|u is any unitarily-invariant norm on Mn(C), if A, N ∈ Mn(C) are polynomially isometric and N is normal, then A is normal. We then extend this result to the infinite-dimensional setting by showing that if A, N ∈ B(H) are polynomially isometric relative to the operator norm and N is a normal operator whose spectrum neither disconnects the plane nor has interior, then A is normal, while if the spectrum of N is not of this form, then there always exists a nonnormal operator B such that B and N are polynomially isometric. Finally, we show that if A and N are compact operators with N normal, and if A and N are polynomially isometric with respect to the (c,p)-norm studied by Chan, Li, and Tu, then A is again normal. [ABSTRACT FROM AUTHOR]

In this paper, we prove that any pair of q-commuting contractions on a Hilbert space dilates to a pair of q-commuting unitaries, where |q| = 1. We generalize this result to a (G, q)-commuting n-tuple (T1, . . . , Tn) of strict contractions, where G is an acyclic graph with vertex set {1, . . . ,n}. We further generalize it to a family of (G, q)-commuting strict contractions, where G is an acyclic graph on an infinite set of vertices. [ABSTRACT FROM AUTHOR]

In this paper, we will show that the Hesselink stratification of a Hilbert scheme of hypersurfaces is independent of the choice of Plücker coordinate and there is a positive relation between the length of Hesselink's worst virtual 1-parameter subgroup and multiplicity of a projective hypersurface. [ABSTRACT FROM AUTHOR]

The spectrum of the singular indefinite Sturm-Liouville operator ... with a real potential q ∊ L¹(ℝ) covers the whole real line, and, in addition, non-real eigenvalues may appear if the potential q assumes negative values. A quantitative analysis of the non-real eigenvalues is a challenging problem, and so far only partial results in this direction have been obtained. In this paper the bound ... on the absolute values of the non-real eigenvalues λ of A is obtained. Furthermore, separate bounds on the imaginary parts and absolute values of these eigenvalues are proved in terms of the L¹-norm of the negative part of q. [ABSTRACT FROM AUTHOR]

Akemann and Weaver (2014) have shown a remarkable extension of Weaver's KSr Conjecture (2004) in the form of approximate Lyapunov's theorem. This was made possible thanks to the breakthrough solution of the Kadison-Singer problem by Marcus, Spielman, and Srivastava (2015). In this paper we show a similar type of Lyapunov's theorem for continuous frames on non-atomic measure spaces. In contrast with discrete frames, the proof of this result does not rely on the recent solution of the Kadison-Singer problem. [ABSTRACT FROM AUTHOR]

This paper extends two recent improvements in the Hilbert space setting of the well-known Katznelson-Tzafriri theorem by establishing both a version of the result valid for bounded representations of a large class of abelian semigroups and a quantified version for contractive representations. The paper concludes with an outline of an improved version of the Katznelson-Tzafriri theorem for individual orbits, whose validity extends even to certain unbounded representations. [ABSTRACT FROM AUTHOR]

In the present paper, we continue our research on numerical ranges of operators. With newly developed techniques, we show that Let N be a nest on a Hilbert space H and Τ ∈ T (N), where T (N) denotes the nest algebra associated with N. Then for given ε > 0, there exists a compact operator K with llKll< ε such that T +K ∈ T (N) and the numerical range of T + K is closed. As applications, we show that the statement of the above type holds for the class of Cowen-Douglas operators, the class of nilpotent operators and the class of quasinilpotent operators. [ABSTRACT FROM AUTHOR]

In this paper, we study the relation between Castelnuovo-Mumford regularity and Bridgeland stability for the Hilbert scheme of n points on P². For the largest n/2 Bridgeland walls, we show that the general ideal sheaf destabilized along a smaller Bridgeland wall has smaller regularity than one destabilized along a larger Bridgeland wall. We give a detailed analysis of the case of monomial schemes and obtain a precise relation between the regularity and the Bridgeland stability for the case of Borel fixed ideals. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the generalized Cauchy-Schwarz inequality for an operator T ∈ L(H) and investigate various properties of operators which satisfy the generalized Cauchy-Schwarz inequality. In particular, every p-hyponormal operator satisfies this inequality. We also prove that if T ∈ L(H) satisfies the generalized Cauchy-Schwarz inequality, then T is paranormal. As an application, we show that if both T and T* in L(H) satisfy the generalized Cauchy-Schwarz inequality, then T is normal. [ABSTRACT FROM AUTHOR]

This paper complements a recent result of Dilworth, Ferenczi, Kutzarova and Odell regarding the ergodicity of strongly asymptotic $ell _p$ spaces. We state this result in a more general form, involving domination relations, and we show that every asymptotically Hilbertian space which is not isomorphic to $ell _2$ is ergodic. In particular, every weak Hilbert space which is not isomorphic to $ell _2$ must be ergodic. Throughout the paper we construct explicitly the maps which establish the fact that the relation $E_0$ is Borel reducible to isomorphism between subspaces of the Banach spaces involved. [ABSTRACT FROM AUTHOR]

The present paper gives a priori upper and lower bounds on nonreal eigenvalues of regular indefinite Sturm-Liouville problems only under the integrability conditions. More generally, a lower bound on non-real eigenvalues of the self-adjoint operator in Krein space is obtained. [ABSTRACT FROM AUTHOR]

If φ is an analytic map of the unit disk D into itself, the composition operator Cφ on a Hardy space H² is defined by Cφ(f) = f o φ. The unilateral shift on H² is the operator of multiplication by z. Beurling (1949) characterized the invariant subspaces for the shift. In this paper, we consider the shift-invariant subspaces that are invariant for composition operators. More specifically, necessary and sufficient conditions are provided for an atomic inner function with a single atom to be invariant for a composition operator, and the Blaschke product invariant subspaces for a composition operator are described. We show that if has Denjoy-Wolff point a on the unit circle, the atomic inner function subspaces with a single atom at a are invariant subspaces for the composition operator Cφ. [ABSTRACT FROM AUTHOR]

Let H1 and H2 be selfadjoint operators or relations (multivalued operators) acting on a separable Hilbert space and assume that the inequality H1 = H2 holds. Then the validity of the inequalities -H-1 1 = -H-12 and H-12 = H-11 is characterized in terms of the inertia of H1 and H2. Such results are known for matrices and boundedly invertible operators. In the present paper those results are extended to selfadjoint, in general unbounded, not necessarily boundedly invertible, operators and, more generally, for selfadjoint relations in separable Hilbert spaces. [ABSTRACT FROM AUTHOR]

The purpose of the present work is to answer an open problem which was raised by G. Costakis and A. Manoussos in their paper "J-class operators and hypercyclicity", J. Operator Theory 67 (2012), 101-119. More precisely, we give the spectral description of the closure of the set of J-class operators acting on a separable Hilbert space. [ABSTRACT FROM AUTHOR]

If X is a compact subset of a Banach space with X -X homogeneous (equivalently 'doubling' or with finite Assouad dimension), then X can be embedded into some R...n (with n sufficiently large) using a linear map L whose inverse is Lipschitz to within logarithmic corrections. More precisely, there exist c, α > 0 such that c ‖x - y‖/| log ‖x - y‖ |α ≤ |Lx - Ly| ≤ c‖x - y‖ for all x, y ∊ X, ‖x - y‖ < δ, for some δ sufficiently small. It is known that one must have α > 1 in the case of a general Banach space and α > 1/2 in the case of a Hilbert space. It is shown in this paper that these exponents can be achieved. While the argument in a general Banach space is relatively straightforward, the Hilbert space case relies on the fact that the maximum volume of a hyperplane slice of a k-fold product of unit volume N-balls is bounded independent of k (this provides a 'qualitative' generalisation of a result on slices of the unit cube due to Hensley (Proc.AMS 73 (1979), 95-100) and Ball (Proc.AMS 97 (1986), 465-473)). [ABSTRACT FROM AUTHOR]

Let B be the ring of bounded operators in a complex, separable Hilbert space. For p > 0 consider the Schatten ideal LP consisting of those operators whose sequence of singular values is p-summable; put S =UpLp Let G be a group and Vcyc the family of virtually cyclic subgroups. Guoliang Yu proved that the K-theory assembly mapH*G(E(G, Vcyc),K(S)) → K*(S[G]) is rationally injective. His proof involves the construction of a certain Chern character tailored to work with coefficients S and the use of some results about algebraic K-theory of operator ideals and about controlled topology and coarse geometry. In this paper we give a different proof of Yu's result. Our proof uses the usual Chern character to cyclic homology. Like Yu's, it relies on results on algebraic K-theory of operator ideals, but no controlled topology or coarse geometry techniques are used. We formulate the result in terms of homotopy K-theory. We prove that the rational assembly map HG*(E(G,Fin),KH(Lp)) ⊗ Q... → KH*(Lp[G]) ⊗ Q... is injective. We show that the latter map is equivalent to the assembly map considered by Yu, and thus obtain his result as a corollary. [ABSTRACT FROM AUTHOR]

For a bounded linear operator T on a complex Hilbert space H, let ΔT,m = Σ m k=0(-1)k ( m k ) T"m-kTm-k for m ∊ N. T is called an misometry if ΔT,m = 0. In this paper, for every even number m, we give an example of invertible (m + 1)-isometry which is not an m-isometry. Next we show that if T is an m-isometry, then the operator ΔT,m-1 is not invertible. [ABSTRACT FROM AUTHOR]

In this paper, we prove that any Hs (Ω) solution u of the problem s (0.1) (-Δ)su = f(u) in Ω,u = 0 on ∂ύ, belongs to L∞ (Ω) for the nonlinearity of f(t) being subcritical and critical. This implies that the solution u is classical if f(t) is C1, γ for some 0 < γ < 1. [ABSTRACT FROM AUTHOR]

An operator T on a complex Hilbert space H is said to be skew symmetric if there exists a conjugate-linear, isometric involution C : H → H so that CTC = --T*. In this paper, we shall give two structure theorems for skew symmetric normal operators. [ABSTRACT FROM AUTHOR]

Let A be an operator algebra on a Hilbert space. We say that an element G ? A is an all-derivable point of A if every derivable linear mapping ? at G (i.e. ?(ST) = ?(S)T + S?(T) for any S, T ∊ algN with ST = G) is a derivation. Suppose that N is a nontrivial complete nest on a Hilbert space H. We show in this paper that G ∊ algN is an all-derivable point if and only if G ≠ 0. [ABSTRACT FROM AUTHOR]

Let D be an integral domain with quotient field K, * be a staroperation on D, and GV *(D) be the set of finitely generated ideals J of D such that J* = D. Then the map *w defined by I*w = {x Є K | Jx ⊆ I for some J Є GV *(D)} for all nonzero fractional ideals I of D is a finite character staroperation on D. In this paper, we study several properties of *w-Noetherian domains. In particular, we prove the Hilbert basis theorem for *w-Noetherian domains. [ABSTRACT FROM AUTHOR]

In this paper we investigate multinormed W*-algebras in terms of the central topologies of W*-algebras. The main result asserts that each multinormed W*-algebra can be realized as a local von Neumann algebra on a certain domain in a Hilbert space. Moreover, it admits the predual (unique up to an isometry), which is the ℓ1-normed space. In the normed case the assertion is reduced to the known Sakai theorem. [ABSTRACT FROM AUTHOR]

In this paper we describe all surjective isometries of the unitary group of a complex Hilbert space. A result on Thompson isometries of the space of all invertible positive elements of a unital C*-algebra is also presented. [ABSTRACT FROM AUTHOR]

An operator T ∊ B(H) is complex symmetric if there exists a conjugate-linear, isometric involution C : H →H so that CTC = T. In this paper, a class of complex symmetric operators on finite dimensional Hilbert spaces is constructed. As an application, it is shown that Kakutani's unilateral weighted shift operator is not complex symmetric; however, it is a norm limit of complex symmetric operators. This gives a negative answer to a question of S. Garcia and W. Wogen: that is, whether or not the class of complex symmetric operators is norm closed. [ABSTRACT FROM AUTHOR]

Unlike Toeplitz operators on H², truncated Toeplitz operators do not have a natural matricial characterization. Consequently, these operators are difficult to study numerically. In this paper we provide criteria for a matrix with distinct eigenvalues to be unitarily equivalent to a truncated Toeplitz operator having an analytic symbol. This test is constructive, and we illustrate it with several examples. As a byproduct, we also prove that every complex symmetric operator on a Hilbert space of dimension ≤ 3 is unitarily equivalent to a direct sum of truncated Toeplitz operators. [ABSTRACT FROM AUTHOR]

In 2006, C. L. Jiang and Z. Y. Wang posed an open problem: If T is a strongly irreducible operator, is A'(T)/rad.A_(T) commutative? They conjectured that the answer is positive. In this paper, to negatively answer their problem, a counterexample is given. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence of almost periodic mild solutions to some second-order equations. Using dichotomy tools and the Schauder fixed point theorem, the existence of almost periodic mild solutions to those second-order evolution equations is established. To illustrate our abstract results, the existence of almost periodic solutions to a damped second-order boundary value problem is also discussed. [ABSTRACT FROM AUTHOR]

In this paper, we show that Beurling's theorem on analytic Hilbert spaces over the complex plane analogous to the Hardy space or the Bergman space does not hold, but for finite co-dimensional quasi-invariant subspaces, they are generated by their wandering subspace if and only if they are generated by $z^n$ provided that the order of the reproducing kernels $K_lambda (z)$ is less than 2 but not equal to 1. [ABSTRACT FROM AUTHOR]

Let $mathfrak {I}$ be a separable Banach ideal in the space of bounded operators acting in a Hilbert space $mathcal {H}$ and $mathcal {U}(mathcal {H})_{mathfrak {I}}$ the Banach-Lie group of unitary operators which are perturbations of the identity by elements in $mathfrak {I}$. In this paper we study the geometry of the unitary orbits $${ U V : U in mathcal {U}(mathcal {H})_{mathfrak {I}}}$$ and $${ U V W^* : U,W in mathcal {U}(mathcal {H})_{mathfrak {I}}},$$ where $V$ is a partial isometry. We give a spatial characterization of these orbits. It turns out that both are included in $V + mathfrak {I}$, and while the first one consists of partial isometries with the same kernel of $V$, the second is given by partial isometries such that their initial projections and $V^*V$ have null index as a pair of projections. We prove that they are smooth submanifolds of the affine Banach space $V + mathfrak {I}$ and homogeneous reductive spaces of $mathcal {U}(mathcal {H})_{mathfrak {I}}$ and $mathcal {U}(mathcal {H})_{mathfrak {I}} times mathcal {U}(mathcal {H})_{mathfrak {I}}$ respectively. Then we endow these orbits with two equivalent Finsler metrics, one provided by the ambient norm of the ideal and the other given by the Banach quotient norm of the Lie algebra of $mathcal {U}(mathcal {H})_{mathfrak {I}}$ (or $mathcal {U}(mathcal {H})_{mathfrak {I}} times mathcal {U}(mathcal {H})_{mathfrak {I}}$) by the Lie algebra of the isotropy group of the natural actions. We show that they are complete metric spaces with the geodesic distance of these metrics. [ABSTRACT FROM AUTHOR]

In this paper we study the asymptotic behaviour under dilations of probability measures supported on polynomial curves in nilmanifolds. We prove, under some mild conditions, the effective equidistribution of such measures to the Haar measure. We also formulate a mean ergodic theorem for $mathbb {R}^n$-representations on Hilbert spaces, restricted to a moving phase of low dimension. Furthermore, we bound the necessary dilation of a given smooth curve in $ mathbb {R}^n$ so that the canonical projection onto $ mathbb {T}^n $ is $ varepsilon $-dense. [ABSTRACT FROM AUTHOR]

Let $H$ be a complex Hilbert space. The symbol $A# B$ stands for the geometric mean of the positive bounded linear operators $A,B$ on $H$ in the sense of Ando. In this paper we describe the general form of all automorphisms of the set of positive operators with respect to the operation $#$. We prove that if $dim Hgeq 2$, any such transformation is implemented by an invertible bounded linear or conjugate-linear operator on $H$. [ABSTRACT FROM AUTHOR]