64 research outputs found
Decomposing numerical ranges along with spectral sets
This note is to indicate the new sphere of applicability of the method
developed by Mlak as well as by the author. Restoring those ideas is summoned
by current developments concerning -spectral sets on numerical ranges
Holomorphic Hermite polynomials in two variables
Generalizations of the Hermite polynomials to many variables and/or to the
complex domain have been located in mathematical and physical literature for
some decades. Polynomials traditionally called complex Hermite ones are mostly
understood as polynomials in and which in fact makes them
polynomials in two real variables with complex coefficients. The present paper
proposes to investigate for the first time holomorphic Hermite polynomials in
two variables. Their algebraic and analytic properties are developed here.
While the algebraic properties do not differ too much for those considered so
far, their analytic features are based on a kind of non-rotational
orthogonality invented by van Eijndhoven and Meyers. Inspired by their
invention we merely follow the idea of Bargmann's seminal paper (1961) giving
explicit construction of reproducing kernel Hilbert spaces based on those
polynomials. "Homotopic" behavior of our new formation culminates in comparing
it to the very classical Bargmann space of two variables on one edge and the
aforementioned Hermite polynomials in and on the other. Unlike in
the case of Bargmann's basis our Hermite polynomials are not product ones but
factorize to it when bonded together with the first case of limit properties
leading both to the Bargmann basis and suitable form of the reproducing kernel.
Also in the second limit we recover standard results obeyed by Hermite
polynomials in and
On oscillatorlike Hamiltonians and squeezing
Generalizing a recent proposal leading to one-parameter families of
Hamiltonians and to new sets of squeezed states, we construct larger classes of
physically admissible Hamiltonians permitting new developments in squeezing.
Coherence is also discussed.Comment: 15 pages, Late
Squeezing: the ups and downs
We present an operator theoretic side of the story of squeezed states
regardless the order of squeezing. For low order, that is for displacement
(order 1) and squeeze (order 2) operators, we bring back to consciousness what
is know or rather what has to be known by making the exposition as exhaustive
as possible. For the order 2 (squeeze) we propose an interesting model of the
Segal-Bargmann type. For higher order the impossibility of squeezing in the
traditional sense is proved rigorously. Nevertheless what we offer is the
state-of-the-art concerning the topic.Comment: 21 pages; improved presentation; it has been published by Proceedings
of the Royal Society
Componentwise and Cartesian decompositions of linear relations
Let be a, not necessarily closed, linear relation in a Hilbert space
\sH with a multivalued part \mul A. An operator in \sH with \ran
B\perp\mul A^{**} is said to be an operator part of when A=B \hplus
(\{0\}\times \mul A), where the sum is componentwise (i.e. span of the
graphs). This decomposition provides a counterpart and an extension for the
notion of closability of (unbounded) operators to the setting of linear
relations. Existence and uniqueness criteria for the existence of an operator
part are established via the so-called canonical decomposition of . In
addition, conditions are developed for the decomposition to be orthogonal
(components defined in orthogonal subspaces of the underlying space). Such
orthogonal decompositions are shown to be valid for several classes of
relations. The relation is said to have a Cartesian decomposition if
A=U+\I V, where and are symmetric relations and the sum is
operatorwise. The connection between a Cartesian decomposition of and the
real and imaginary parts of is investigated
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