Second-order operators with degenerate coefficients

We consider properties of second-order operators $H = -\sum^d_{i,j=1} \partial_i \, c_{ij} \, \partial_j$ on \Ri^d with bounded real symmetric measurable coefficients. We assume that $C = (c_{ij}) \geq 0$ almost everywhere, but allow for the possibility that $C$ is singular. We associate with $H$ a canonical self-adjoint viscosity operator $H_0$ and examine properties of the viscosity semigroup $S^{(0)}$ generated by $H_0$. The semigroup extends to a positive contraction semigroup on the $L_p$-spaces with $p \in [1,\infty]$. We establish that it conserves probability, satisfies $L_2$~off-diagonal bounds and that the wave equation associated with $H_0$ has finite speed of propagation. Nevertheless $S^{(0)}$ is not always strictly positive because separation of the system can occur even for subelliptic operators. This demonstrates that subelliptic semigroups are not ergodic in general and their kernels are neither strictly positive nor H\"older continuous. In particular one can construct examples for which both upper and lower Gaussian bounds fail even with coefficients in C^{2-\varepsilon}(\Ri^d) with $\varepsilon > 0$.Comment: 44 page

Cesaro bounded operators in Banach spaces

[EN] We study several notions of boundedness for operators. It is known that any power bounded operator is absolutely Cesaro bounded and strongly Kreiss bounded (in particular, uniformly Kreiss bounded). The converses do not hold in general. In this note, we give examples of topologically mixing (hence, not power bounded) absolutely Cesaro bounded operators on l(p)(N), 1 <= p < infinity, and provide examples of uniformly Kreiss bounded operators which are not absolutely Cesaro bounded. These results complement a few known examples (see [27] and [2]). We also obtain a characterization of power bounded operators which generalizes a result of Van Casteren [32]. In [2] Aleman and Suciu asked if every uniformly Kreiss bounded operator T on a Banach space satisfies that. We solve this question for Hilbert space operators and, moreover, we prove that, if T is absolutely Cesaro bounded on a Banach (Hilbert) space, then parallel to T-n parallel to = o(n) ((parallel to Tn parallel to=o(n12), respectively). As a consequence, every absolutely Cesaro bounded operator on a reflexive Banach space is mean ergodic.The first, second and fourth authors were supported by MINECO and FEDER, Project MTM201675963-P. The third author was supported by grant No. 17-27844S of GA CR and RVO: 67985840. The fourth author was also supported by Generalitat Valenciana, Project PROMETEO/2017/102.Bermúdez, T.; Bonilla, A.; Muller, V.; Peris Manguillot, A. (2020). Cesaro bounded operators in Banach spaces. Journal d Analyse Mathématique. 140(1):187-206. https://doi.org/10.1007/s11854-020-0085-8S1872061401A. Albanese, J. Bonet and W. J. Ricker, Mean ergodic operators in Fréchet spaces, Ann. Acad. Sci. Fenn. Math. 34 (2009), 401–436.A. Aleman and L. Suciu, On ergodic operator means in Banach spaces, Integral Equations Operator Theory 85 (2016), 259–287.I. Assani, Sur les opérateurs à puissances bornées et le théorème ergodique ponctuel dans Lp[0, 1], 1 < p < ∞, Canad. J. Math. 38 (1986), 937–946.M. J. Beltrán-Meneu, Operators on Weighted Spaces of Holomorphic Functions, PhD Thesis, Universitat Politècnica de Valencia, Valencia, Spain, 2014.M. J. Beltrán, J. Bonet and C. Fernández, Classical operators on weighted Banach spaces of entire functions, Proc. Amer. Math. Soc. 141 (2013), 4293–4303.M. J. Beltrán, M.C. Gómez-Collado, E. Jordá and D. Jornet, Mean ergodic composition operators on Banach spaces of holomorphic functions, J. Funct. Anal. 270 (2016), 4369–4385.N. C. Bernardes, Jr., A. Bonilla, V. Müller and A. Peris, Distributional chaos for linear operators, J. Funct. Anal. 265 (2013), 2143–2163.N. C. Bernardes, Jr., A. Bonilla, A. Peris and X. Wu, Distributional chaos for operators in Banach spaces, J. Math. Anal. Appl. 459 (2018), 797–821.J. Bonet, Dynamics of the differentiation operator on weighted spaces of entire functions, Math. Z. 261 (2009), 649–657.Y. Derriennic, On the mean ergodic theorem for Cesaro bounded operators, Colloq. Math. 84/85 (2000), 443–455.Y. Derriennic and M. Lin, On invariant measures and ergodic theorems for positive operators, J. Funct. Anal. 13 (1973), 252–267.R. Émilion, Mean-bounded operators and mean ergodic theorems, J. Funct. Anal. 61 (1985), 1–14.A. Gomilko and J. Zemánek, On the uniform Kreiss resolvent condition, (Russian) Funktsional. Anal. i Prilozhen. 42 (2008), 81–84A. Gomilko and J. Zemánek, English translation in Funct. Anal. Appl. 42 (2008), 230–233.K.-G. Grosse-Erdmann and A. Peris, Linear Chaos, Springer, London, 2011.B. Z. Guo and H. Zwart, On the relation between stability of continuous- and discrete-time evolution equations via the Cayley transform, Integral Equations Operator Theory 54 (2006), 349–383.B. Hou and L. Luo, Some remarks on distributional chaos for bounded linear operators, Turk. J. Math. 39 (2015), 251–258.E. Hille, Remarks on ergodic theorems, Trans. Amer. Math. Soc. 57 (1945), 246–269.I. Kornfeld and W. Kosek, Positive L1operators associated with nonsingular mappings and an example of E. Hille, Colloq. Math. 98 (2003), 63–77.W. Kosek, Example of a mean ergodic L1operator with the linear rate of growth, Colloq. Math. 124 (2011), 15–22.U. Krengel, Ergodic Theorems, Walter de Gruyter, Berlin, 1985.C. Lubich and O. Nevanlinna, On resolvent conditions and stability estimates, BIT, 31 (1991), 293–313.C. A. McCarthy, A Strong Resolvent Condition does not Imply Power-Boundedness, Chalmers Institute of Technology and the University of Göteborg, Preprint No. 15 (1971).A. Montes-Rodríguez, J. Sánchez-Álvarez and J. Zemánek, Uniform Abel—Kreiss boundedness and the extremal behavior of the Volterra operator, Proc. London Math. Soc. 91 (2005), 761–788.V. Müller and J. Vrsovsky, Orbits of linear operators tending to infinity, Rocky Mountain J. Math. 39 (2009), 219–230.O. Nevanlinna, Resolvent conditions and powers of operators, Studia Math. 145 (2001), 113–134.J. Neveu, Mathematical Foundations of the Calculus of Probability, Holden-Day, San Francisco, Calif.-London-Amsterdam, 1965.A. L. Shields, On Möbius Bounded operators, Acta Sci. Math. (Szeged) 40 (1978), 371–374.J. C. Strikwerda and B. A. Wade, A survey of the Kreiss matrix theorem for power bounded families of matrices and its extensions, in Linear Operators, Polish Acad. Sci., Warsaw, 1997, pp. 339–360.L. Suciu, Ergodic behaviors of the regular operator means, Banach J. Math. Anal. 11 (2017), 239–265.L. Suciu and J. Zemánek, Growth conditions on Cesàro means of higher order, Acta Sci. Math (Szeged) 79 (2013), 545–581.Y. Tomilov and J. Zemánek, A new way of constructing examples in operator ergodic theory, Math. Proc. Cambridge Philos. Soc. 137 (2004), 209–225.J. A. Van Casteren, Boundedness properties of resolvents and semigroups of operators, in Linear Operators, Polish Acad. Sci. Inst. Math., Warsaw, 1997, pp. 59–74

Dynamics of Totally Constrained Systems II. Quantum Theory

In this paper a new formulation of quantum dynamics of totally constrained systems is developed, in which physical quantities representing time are included as observables. In this formulation the hamiltonian constraints are imposed on a relative probability amplitude functional $\Psi$ which determines the relative probability for each state to be observed, instead of on the state vectors as in the conventional Dirac quantization. This leads to a foliation of the state space by linear manifolds on each of which $\Psi$ is constant, and dynamics is described as linear mappings among acausal subspaces which are transversal to these linear manifolds. This is a quantum analogue of the classical statistical dynamics of totally constrained systems developed in the previous paper. It is shown that if the von Neumann algebra \C generated by the constant of motion is of type I, $\Psi$ can be consistently normalizable on the acausal subspaces on which a factor subalgebra of \C is represented irreducibly, and the mappings among these acausal subspaces are conformal. How the formulation works is illustrated by simple totally constrained systems with a single constraint such as the parametrized quantum mechanics, a relativistic free particle in Minkowski and curved spacetimes, and a simple minisuperspace model. It is pointed out that the inner product of the relative probability amplitudes induced from the original Hilbert space picks up a special decomposition of the wave functions to the positive and the negative frequency modes.Comment: 57 pages, some unexpected control codes in the original file, which may cause errors for some LaTeX compilers, were remove

Monotone and Boolean Convolutions for Non-compactly Supported Probability Measures

The equivalence of the characteristic function approach and the probabilistic approach to monotone and boolean convolutions is proven for non-compactly supported probability measures. A probabilistically motivated definition of the multiplicative boolean convolution of probability measures on the positive half-line is proposed. Unlike Bercovici's multiplicative boolean convolution it is always defined, but it turns out to be neither commutative nor associative. Finally some relations between free, monotone, and boolean convolutions are discussed.Comment: 32 pages, new Lemma 2.