Phone: 618-650-5070
Email: jloreau@siue.edu
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I study operator theory, whose primary objects consist of operators (i.e., continuous linear transformations) on a Hilbert space $\mathcal {H}$ (i.e., a complete inner product space). When $\mathcal {H}$ is finite dimensional, it is isomorphic to $\mathbb {C}^n$ and hence these transformations are naturally identified with the $n\times n$ matrices $M_n(\mathbb {C})$ by the choice of an orthonormal basis $\mathfrak {e}=\{e_1,\ldots ,e_n\}$ for $\mathcal {H}$. Given an operator $T$ and an orthonormal basis $\mathfrak {e}$ the associated matrix representation $[T]_{\mathfrak {e}}$ has a diagonal sequence. As $\mathfrak {e}$ ranges over all orthonormal bases for $\mathcal {H}$ we can examine all possible diagonal sequences associated with $T$. When $\mathcal {H}$ is separable and infinite dimensional (i.e, has a countably infinite orthonormal basis), we can similarly define the infinite diagonal sequences associated to an operator $T$.
My current research concerns the interplay between operators and their diagonal sequences. Such work has a long history in operator theory, especially concerning selfadjoint operators (i.e., those whose matrices are equal to their conjugate transpose). The combined work of [Sch23] and [Hor54] characterizes the diagonal sequences of selfadjoint operators on a finite dimensional space in terms of their eigenvalues and a preorder on finite real-valued sequences called majorization. Precisely, the classical Schur–Horn Theorem states that the sequence $d = \langle d_1,\ldots ,d_n \rangle $ appears as a diagonal of a selfadjoint operator with eigenvalues $\lambda = \langle \lambda _1,\ldots ,\lambda _n \rangle $ if and only if their nonincreasing rearrangements $d^*,\lambda ^*$ satisfy \[ \sum _{i=1}^k d^*_i \le \sum _{i=1}^k \lambda ^*_i, \quad \text{for}\ 1 \le k \le n \] with equality when $k=n$. The inequalities specified above are collectively called majorization.
Recent work by Arveson–Kadison, Kaftal–Weiss, Bownik–Jasper, among others, has focused on extending the Schur–Horn theorem to the infinite dimensional setting ([Mar64], [Neu99], [AK06], [KW10]; [Kad02a; Kad02b], [Jas13], [BJ13]). My work with G. Weiss in [LW14b] concerns a Schur–Horn theorem for positive compact operators. The important feature of positive compact operators is that each has a nonnegative eigenvalue sequence $\lambda $ which converges to zero. When $\lambda $ does not contain zero, it admits a nonincreasing rearrangement and so also does each of the operator’s diagonal sequences. Majorization naturally extends to this situation and, indeed, [KW10] proves a Schur–Horn theorem in this scenario. The case when $\lambda $ does contain zero is significantly harder since $\lambda $ does not admit a nonincreasing rearrangement, and so a new kind of majorization is necessary. In [LW14b] we define and investigate variations of majorization and use them to obtain a complete characterization of diagonal sequences when $\lambda $ contains zero with infinite multiplicity, and a partial characterization of the diagonal sequences when zero has finite multiplicity. One of my current research goals is to solve in its entirety this latter case of finite multiplicity.
In recent decades we have also seen a push to investigate diagonals of certain non-selfadjoint operators by Fan–Fong–Herrero, Giol–Kovalev–Larson–Nguyen–Tener and others ([Fan84], [FFH87], [FF94], [Fon86]). Prompted by a question of J. Jasper we provide a characterization in [LW14a] of diagonals of idempotent operators (i.e., $T^2=T$). This extends the results of [Kad02a; Kad02b] where the idempotents are projections (i.e., selfadjoint idempotents), and [GKLNT11] where the idempotents are $n\times n$ matrices. One particularly interesting fact we discovered is that a nonzero idempotent operator has the constant zero sequence as a diagonal if and only if it is not a Hilbert–Schmidt perturbation of a projection, that is, if the idempotent $T$ cannot be expressed as $P+K$ where $P$ is a projection and $K$ is an operator whose matrix representations have square-summable entries. The existence of idempotents with zero-diagonal in infinite dimensions is in sharp contrast to the finite dimensional case where the sum of the diagonal entries of an idempotent is always equal to its rank, hence is positive when $T\not =0$. This shows how we can elicit information about an operator from its diagonal sequences; the existence of the zero-sequence for an idempotent $T$ forces $T$ to be sufficiently far from the set of projections in an appropriate sense.
A type II$_{1}$ factor can be thought of as an analogue of $M_n(\mathbb {C})$ with ‘continuous dimensionality’. All of this may be extended to type $\mathrm {II}_1$ factors via the following translation. Replace the space of bounded operators by a $\mathrm {II}_1$ factor $\mathcal {M}$, and let a maximal abelian selfadjoint subalgebra (masa) $\mathcal {A}\subseteq \mathcal {M}$ take the place of the space of diagonal sequences with respect to a basis $\mathfrak {e}$, and let the trace-preserving conditional expectation $E_{\mathcal {A}}:\mathcal {M}\to \mathcal {A}$ replace the map that associates to an operator its ‘diagonal’. In this situation there is already an appropriate notion of majorization and so it is natural to ask about Schur-Horn theorems. This has been considered by [Rav12], [AM07], [AM08] and others. Part of my current interest lies in obtaining the strongest possible versions of these results, as we did for positive compact operators in [LW14b].
I have ongoing projects with Gary Weiss and John Jasper, Victor Kaftal, Catalin Dragan and Dewey Estep. My work with John and Gary concerns infinite dimensional versions of Thompson’s Theorem [Tho77] which characterizes diagonals of the set of operators with prescribed singular values (eigenvalues of $\sqrt {T^{*}T}$). We extend Thompson’s Theorem to general compact operators and then go on to consider diagonals of unitary operators. The latter problem was investigated in finite dimensions by Horn [Hor54]. Along with Victor Kaftal I am providing new insight into Kadison’s generalized Pythagorean Theorem [Kad02a; Kad02b] through the lens of essential codimension which arose from the theory of Brown, Douglas and Fillmore [BDF73]. My work with Dewey Estep concerns the existence of compact subsets of the plane equipped with the path metric which have nonunique geodesics between every pair of points.
I am also interested in the question of what conditions on a masa $\mathcal {A}\subseteq \mathcal {M}$ guarantee the existence of another masa $\mathcal {B}$ such that the range of the trace-preserving conditional expectation $E_{\mathcal {A}}(\mathcal {B})$ consists entirely of scalars. In some sense, this should say that the masa $\mathcal {A}$ contains no information about the masa $\mathcal {B}$. Although the analogous result has a straightforward solution for matrices $M_n(\mathbb {C})$ and the space of bounded operators in general, the answer is not quite as obvious for type II factors.
It is often thought that there are no more interesting questions left to ask in linear algebra, but this is far from the truth; there is much we do not understand about $M_n(\mathbb {C})$. For example, another project on which I am working is determining whether or not diagonal sequences of an operator $T\in M_n(\mathbb {C})$ completely characterize the unitary orbit $\mathcal {U}(T)=\{UTU^{*} \mid U\ \text{unitary} \}$. If they do, I will investigate to what extent this is true for bounded operators in general. These problems are characteristic of the overarching theme of my research in that my interests pertain primarily to problems which have analogues in $M_n(\mathbb {C})$ and to the development of extensions to infinite dimensions and von Neumann algebras. In this sense, I am curious about how far-reaching are the results of linear algebra.
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, Feb. 2013.
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J. Loreaux and G. Weiss, Diagonality and idempotents with appplications
to problems in operator theory and frame theory, submitted to Journal of
Operator Theory, arXiv: 1410.7441 [math.FA]
, Oct. 2014.
J. Loreaux and G. Weiss, Majorization and a Schur–Horn Theorem for
positive compact operators, the nonzero kernel case, Journal of Functional
Analysis (Mar. 2014), arXiv: 1403.4917 [math.FA]
, to appear.
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