On power bounded operators with holomorphic eigenvectors, II

In [U] (among other results), M. Uchiyama gave the necessary and sufficient conditions for contractions to be similar to the unilateral shift $S$ of multiplicity $1$ in terms of norm-estimates of complete analytic families of eigenvectors of their adjoints. In [G2], it was shown that this result for contractions can't be extended to power bounded operators. Namely, a cyclic power bounded operator was constructed which has the requested norm-estimates, is a quasiaffine transform of $S$, but is not quasisimilar to $S$. In this paper, it is shown that the additional assumption on a power bounded operator to be quasisimilar to $S$ (with the requested norm-estimates) does not imply similarity to $S$. A question whether the criterion for contractions to be similar to $S$ can be generalized to polynomially bounded operators remains open. Also, for every cardinal number $2\leq N\leq \infty$ a power bounded operator $T$ is constructed such that $T$ is a quasiaffine transform of $S$ and $\dim\ker T^*=N$. This is impossible for polynomially bounded operators. Moreover, the constructed operators $T$ have the requested norm-estimates of complete analytic families of eigenvectors of $T^*$

Technological Progress and the Distribution of Productivities across Sectors

This paper studies the impact of the process of technological change on the distribution of productivities and profits across sectors. We find that if technological progress affects high-tech and traditional sectors differently, the impact of changes in the determinants of economic growth may differ depending on which is the actual change. When an economy is growing faster due to an increase in the productivity of research or to a reduction of the taxes on capital accumulation, inequality will decrease. However, if faster growth is due to the presence of tax incentives to high technology sectors or to structural changes that allow a better absorption of externalities, inequality will increase.

Harnack inequalities in infinite dimensions

We consider the Harnack inequality for harmonic functions with respect to three types of infinite dimensional operators. For the infinite dimensional Laplacian, we show no Harnack inequality is possible. We also show that the Harnack inequality fails for a large class of Ornstein-Uhlenbeck processes, although functions that are harmonic with respect to these processes do satisfy an a priori modulus of continuity. Many of these processes also have a coupling property. The third type of operator considered is the infinite dimensional analog of operators in H\"{o}rmander's form. In this case a Harnack inequality does hold.Comment: Minor revision of the previous versio