Rota’s universal operators and invariant subspaces in Hilbert spaces

A Hilbert space operator is called universal (in the sense of Rota) if every operator on the Hilbert space is similar to a multiple of the restriction of the universal operator to one of its invariant subspaces. We exhibit an analytic Toeplitz operator whose adjoint is universal in the sense of Rota and commutes with a quasi-nilpotent injective compact operator with dense range. In articular, this new universal operator invites an approach to the Invariant Subspace Problem that uses properties of operators that commute with the universal operator.

Mackey topology on locally convex spaces and on locally quasi-convex groups. Similarities and historical remarks

A counterpart of the Mackey–Arens Theorem for the class of locally quasi-convex topological Abelian groups (LQC-groups) was initiated in Chasco et al. (Stud Math 132(3):257–284, 1999). Several authors have been interested in the problems posed there and have done clarifying contributions, although the main question of that source remains open. Some differences between the Mackey Theory for locally convex spaces and for locally quasi-convex groups, stem from the following fact: The supremum of all compatible locally quasi-convex topologies for a topological abelian group G may not coincide with the topology of uniform convergence on the weak quasi-convex compact subsets of the dual groupG∧. Thus, a substantial part of the classical Mackey–Arens Theorem cannot be generalized to LQC-groups. Furthermore, the mentioned fact gives rise to a grading in the property of “being a Mackey group”, as defined and thoroughly studied in Díaz Nieto and Martín-Peinador (Proceedings in Mathematics and Statistics 80:119–144, 2014). At present it is not known—and this is the main open question—if the supremum of all the compatible locally quasi-convex topologies on a topological group is in fact a compatible topology. In the present paper we do a sort of historical review on the Mackey Theory, and we compare it in the two settings of locally convex spaces and of locally quasi-convex groups. We point out some general questions which are still open, under the name of Problems.

Rota's universal operators and invariant subspaces in Hilbert spaces

A Hilbert space operator is called universal (in the sense of Rota) if every operator on the Hilbert space is similar to a multiple of the restriction of the universal operator to one of its invariant subspaces. We exhibit an analytic Toeplitz operator whose adjoint is universal in the sense of Rota and commutes with a quasi-nilpotent injective compact operator with dense range. In particular, this new universal operator invites an approach to the Invariant Subspace Problem that uses properties of operators that commute with the universal operator.

Characterizations of logarithmic Besov spaces in terms of differences, Fourier-analytical decompositions, wavelets and semi-groups.

We work with Besov spaces Bp,q0,b defined by means of differences, with zero classical smoothness and logarithmic smoothness with exponent b. We characterize Bp,q0,b by means of Fourier-analytical decompositions, wavelets and semi-groups. We also compare those results with the well-known characterizations for classical Besov spaces Bp,qs.

On paracompactness, completeness and boundedness in uniform spaces

Recently two new types of completeness in metric spaces, called Bourbaki-completeness and cofinal Bourbaki-completeness, have been introduced in [7]. The purpose of this note is to analyze these completeness properties in the general context of uniform spaces. More precisely, we are interested in how they are related with uniform paracompactness properties, as well as with some kind of uniform boundedness.

Two classes of metric spaces

The class of metric spaces (X,d) known as small-determined spaces, introduced by Garrido and Jaramillo, are properly defined by means of some type of real-valued Lipschitz functions on X. On the other hand, B-simple metric spaces introduced by Hejcman are defined in terms of some kind of bornologies of bounded subsets of X. In this note we present a common framework where both classes of metric spaces can be studied which allows us to see not only the relationships between them but also to obtain new internal characterizations of these metric properties.

Caught in the act: gas and stellar velocity dispersions in a fast Quenching compact star-forming galaxy at z ~ 1.7

We present Keck I MOSFIRE spectroscopy in the Y and H bands of GDN-8231, a massive, compact, star-forming galaxy at a redshift of z ~ 1.7. Its spectrum reveals both Hα and [Nii] emission lines and strong Balmer absorption lines. The Hα and Spitzer MIPS 24 μm fluxes are both weak, thus indicating a low star-formation rate of SFR≲5-10 M_⨀ yr−1. This, added to a relatively young age of ~700 Myr measured from the absorption lines, provides the first direct evidence for a distant galaxy being caught in the act of rapidly shutting down its star formation. Such quenching allows GDN-8231 to become a compact, quiescent galaxy, similar to three other galaxies in our sample, by z ~ 1.5. Moreover, the color profile of GDN-8231 shows a bluer center, consistent with the predictions of recent simulations for an early phase of inside-out quenching. Its line-of-sight velocity dispersion for the gas, σ_LOG^gas = 127 ± 32 km s^−1, is nearly 40% smaller than that of its stars, σ_LOG^* = 215 ± 35 km s^−1. High-resolution hydro-simulations of galaxies explain such apparently colder gas kinematics of up to a factor of ~1.5 with rotating disks being viewed at different inclinations and/or centrally concentrated star-forming regions. A clear prediction is that their compact, quiescent descendants preserve some remnant rotation from their star-forming progenitors.

On the nullstellensätze for stein spaces and C-analytic sets.

In this work we prove the real Nullstellensatz for the ring O(X) of analytic functions on a C-analytic set X ⊂ Rn in terms of the saturation of Łojasiewicz’s radical in O(X): The ideal I(Ƶ(a)) of the zero-set Ƶ(a) of an ideal a of O(X) coincides with the saturation (Formula presented) of Łojasiewicz’s radical (Formula presented). If Ƶ(a) has ‘good properties’ concerning Hilbert’s 17th Problem, then I(Ƶ(a)) = (Formula presented) where (Formula presented) stands for the real radical of a. The same holds if we replace (Formula presented) with the real-analytic radical (Formula presented) of a, which is a natural generalization of the real radical ideal in the C-analytic setting. We revisit the classical results concerning (Hilbert’s) Nullstellensatz in the framework of (complex) Stein spaces. Let a be a saturated ideal of O(Rn) and YRn the germ of the support of the coherent sheaf that extends aORn to a suitable complex open neighborhood of Rn. We study the relationship between a normal primary decomposition of a and the decomposition of YRn as the union of its irreducible components. If a:= p is prime, then I(Ƶ(p)) = p if and only if the (complex) dimension of YRn coincides with the (real) dimension of Ƶ(p).

CB-norm estimates for maps between noncommutative Lp-spaces and quantum channel theory.

In the first part of this work, we show how certain techniques from quantum information theory can be used in order to obtain very sharp embeddings between noncommutative Lp-spaces. Then, we use these estimates to study the classical capacity with restricted assisted entanglement of the quantum erasure channel and the quantum depolarizing channel. In particular, we exactly compute the capacity of the first one and we show that certain nonmultiplicative results hold for the second one.

CANDELS: The progenitors of compact quiescent galaxies at z ~ 2

We combine high-resolution Hubble Space Telescope/WFC3 images with multi-wavelength photometry to track the evolution of structure and activity of massive (M_*> 10^10 M_☉) galaxies at redshifts z = 1.4-3 in two fields of the Cosmic Assembly Near-infrared Deep Extragalactic Legacy Survey. We detect compact, star-forming galaxies (cSFGs) whose number densities, masses, sizes, and star formation rates (SFRs) qualify them as likely progenitors of compact, quiescent, massive galaxies (cQGs) at z = 1.5-3. At z≲2, cSFGs present SFR = 100-200 M_☉ yr^–1, yet their specific star formation rates (sSFR ~ 10^–9 yr^–1) are typically half that of other massive SFGs at the same epoch, and host X-ray luminous active galactic nuclei (AGNs) 30 times (~30%) more frequently. These properties suggest that cSFGs are formed by gas-rich processes (mergers or disk-instabilities) that induce a compact starburst and feed an AGN, which, in turn, quench the star formation on dynamical timescales (few 10^8 yr). The cSFGs are continuously being formed at z = 2-3 and fade to cQGs down to z ~ 1.5. After this epoch, cSFGs are rare, thereby truncating the formation of new cQGs. Meanwhile, down to z = 1, existing cQGs continue to enlarge to match local QGs in size, while less-gas-rich mergers and other secular mechanisms shepherd (larger) SFGs as later arrivals to the red sequence. In summary, we propose two evolutionary tracks of QG formation: an early (z≲2), formation path of rapidly quenched cSFGs fading into cQGs that later enlarge within the quiescent phase, and a late-arrival (z≳2) path in which larger SFGs form extended QGs without passing through a compact state.