Fredholm operators and essential S-spectrum in the quaternionic setting

As earlier conjectured by several authors and much later established by Solèr, from the lattice-theory point of view, Quantum Mechanics may be formulated in real, complex or quaternionic Hilbert spaces only. On the other hand, no quantum systems seem to exist that are naturally described in a real or quaternionic Hilbert space. In a previous paper [23], we showed that any quantum system which is elementary from the viewpoint of the Poincaré symmetry group and it is initially described in a real Hilbert space, it can also be described within the standard complex Hilbert space framework. This complex description is unique and more precise than the real one as, for instance, in the complex description, all self-adjoint operators represent observables defined by the symmetry group. The complex picture fulfils the thesis of Solér’s theorem and permits the standard formulation of the quantum Noether’s theorem. The present work is devoted to investigate the remaining case, namely, the possibility of a description of a relativistic elementary quantum system in a quaternionic Hilbert space. Everything is done exploiting recent results of the quaternionic spectral theory that were independently developed. In the initial part of this work, we extend some results of group representation theory and von Neumann algebra theory from the real and complex cases to the quaternionic Hilbert space case. We prove the double commutant theorem also for quaternionic von Neumann algebras (whose proof requires a different procedure with respect to the real and complex cases) and we extend to the quaternionic case a result established in the previous paper concerning the classification of irreducible von Neumann algebras into three categories. In the second part of the paper, we consider an elementary relativistic system within Wigner’s approach defined as a locally-faithful irreducible strongly-continuous unitary representation of the Poincaré group in a quaternionic Hilbert space. We prove that, if the squared-mass operator is non-negative, the system admits a natural, Poincaré invariant and unique up to sign, complex structure which commutes with the whole algebra of observables generated by the representation itself. This complex structure leads to a physically equivalent reformulation of the theory in a complex Hilbert space. Within this complex formulation, differently from what happens in the quaternionic one, all self-adjoint operators represent observables in agreement with Solèr’s thesis, the standard quantum version of Noether theorem may be formulated and the notion of composite system may be given in terms of tensor product of elementary systems. In the third part of the paper, we focus on the physical hypotheses adopted to define a quantum elementary relativistic system relaxing them on the one hand, and making our model physically more general on the other hand. We use a physically more accurate notion of irreducibility regarding the algebra of observables only, we describe the symmetries in terms of automorphisms of the restricted lattice of elementary propositions of the quantum system and we adopt a notion of continuity referred to the states viewed as probability measures on the elementary propositions. Also in this case, the final result proves that there exists a unique (up to sign) Poincaré invariant complex structure making the theory complex and completely fitting into Solèr’s picture. The overall conclusion is that relativistic elementary systems are naturally and better described in complex Hilbert spaces even if starting from a real or quaternionic Hilbert space formulation and this complex description is uniquely fixed by physics.

A representation of Weyl–Heisenberg Lie algebra in the quaternionic setting

From the viewpoint of the theory of orthomodular lattices of elementary propositions, Quantum Theories can be formulated in real, complex or quaternionic Hilbert spaces as established in Sol\'er's theorem. The said lattice eventually coincides with the lattice of all orthogonal projectors on a separable Hilbert space over R, C, or over the algebra of quaternions H. Quantum states are $\sigma$-additive probability measures on that non-Boolean lattice. Gleason's theorem proves that, if the Hilbert space is separable with dimension >2 and the Hilbert space is either real or complex, then states are one-to-one with standard density matrices (self-adjoint, positive, unit-trace, trace-class operators). The extension of this result to quaternionic Hilbert spaces was obtained by Varadarajan in 1968. Unfortunately, even if the hard part of the proof is correct, the formulation of this extension is mathematically incorrect. This is due to some peculiarities of the notion of trace in quaternionic Hilbert spaces, e.g., basis dependence, making the theory of trace-class operators in quaternionic Hilbert spaces different from the standard theory in real and complex Hilbert spaces. A minor issue also affects Varadarajan's statement for real Hilbert space formulation. This paper is mainly devoted to present Gleason-Varadarajan's theorem into a technically correct form valid for the three types of Hilbert spaces. After having develped part of the general mathematical technology of trace-class operators in (generally non-separable) quaternionic Hilbert spaces, we prove that only the {\em real part} of the trace enters the formalism of quantum theories (also dealing with unbounded observables and symmetries) and it can be safely used to formulate and prove a common statement of Gleason's theorem.

Using a left multiplication defined on a right quaternionic Hilbert space, we shall demonstrate that various classes of coherent states such as the canonical coherent states, pure squeezed states which are obtained by the sole action of the squeeze operator on the vacuum state, fermionic coherent states can be defined with all the desired properties on a right quaternionic Hilbert space. Furthermore, we shall also demonstrate that squeezed states, which are obtained by the action of the squeeze operator on canonical coherent states, in other words, they are obtained by the action of the squeeze operator followed by the action of the displacement operator on the vacuum state, can be defined on the same Hilbert space, but the noncommutativity of quaternions prevents us in getting the desired results.

A continuous real-linear operator A = A0 + i1A1 + i2A2 + i3A3 on a quaternionic Hilbert space is called sesquihermitian if the linear operators Av are Hermitian; this condition is independent of the choice of quaternion basis (i1,i2,i3). The joint spectral distribution of the Av provides a functional calculus for sesquihermitian operators and real-valued C∞-functions on . This calculus is independent of the quaternion basis and extends naturally to quaternion-valued functions to give a continuous quaternion-linear mapping from the algebra of these functions to that of sesquihermitian operators. The mapping is not, in general, multiplicative unless the Av commute, in which case it agrees with that for several commuting operators on complex Hilbert space.

The numerical range in the quaternionic setting is, in general, a non-convex subset of the quaternions. The essential numerical range is a refinement of the numerical range that only keeps the elements that have, in a certain sense, infinite multiplicity. We prove that the essential numerical range of a bounded linear operator on a quaternionic Hilbert space is convex. A quaternionic analogue of Lancaster theorem, relating the closure of the numerical range and its essential numerical range, is also provided.

In this paper, we introduce and study the frames in separable quaternionic Hilbert spaces. Results on the existence of frames in quaternionic Hilbert spaces have been given. Also, a characterization of frame in quaternionic Hilbert spaces in terms of frame operator is given. Finally, a Paley-Wiener type perturbation result for frames in quaternionic Hilbert space has been obtained.

Semilinear operators on a complex Hilbert space are studied in a part of a program that aims to develop the theories of additive operators on complex and quaternionic Hilbert spaces for application to problems in mathematical physics. The more notable among the new results proved on the eigenvalue problem for semilinear operators are the following: (i) if α is an eigenvalue of a semilinear operator then so also is any complex number which has the same modulus as α; (ii) if a normal semilinear operator has two eigenvectors belonging to different eigenvalues, then either the two eigenvectors are orthogonal or two eigenvalues have the same moduli; and (iii) a normal semilinear operator has a complete set of eigenvectors if and only if it is self-adjoint. Further, it is shown that there exists a norm-preserving semilinear isomorphism between the spaces of bounded linear and semilinear operators on a complex Hilbert space. Finally it is demonstrated how the theory of semilinear operators can be exploited to solve the problems of finding three involutive mutually anticommuting self-adjoint two-by-two matrices and four four-by-four matrices with the same properties: the unusual and remarkably easy solution of this old familiar exercise establishes the relevance of the theory being developed here to physics.

Introduction. In applying symmetry considerations to quantum mechanics, one is often forced to consider representations T of a symmetry group G by means of Hilbert space operators which may be either unitary or conjugate-unitary. Indeed, in some situations there is a well-determined subgroup N of G (necessarily normal and of index 2 in G) such that T, must be unitary for x E N and conjugate-unitary for x E G N. Such a representation T will be called a conjugating representation of G (relative to N). More generally one can consider projective conjugating representations T, in which the homomorphism relation TT, = T, is replaced by Tj, = A(x, y)Txy (the A(x, y) being complex scalars). These also are required in the applications to quantum mechanics. In ?1, by a small modification of Mackey's analysis, we shall classify the equivalence classes of irreducible projective conjugating representations of G in terms of the irreducible projective representations of N (provided N is of Type I). As an immediate application of this, we obtain in ?2 a classification of all the irreducible unitary representations of any Type I group acting in real or quaternionic Hilbert space, in terms of those acting in complex Hilbert space. In working out this classification for a given group G, the essential step is to subdivide the self-conjugate irreducible complex representations T of G into two classes-those of real type and those of quaternionic type-according as the conjugate-linear map setting up the equivalence of T with its conjugate has positive or negative square. Exactly the same results are obtained if, instead of unitary (not necessarily finite-dimensional) representations, we consider finite-dimensional (not necessarily unitary) representations. The remaining sections of this paper concern finite-dimensional representations of semisimple groups. Fix a connected semisimple Lie group G. The family G of all finite-dimensional (not necessarily unitary) irreducible complex representations of G is parametrized by the well-known Cartan-Weyl method of dominant weights. Which of the self-conjugate elements of Gi are of real type and which are of quaternionic type? In ??3 and 4 this question is answered in two steps: First, Theorem 5 reduces the general case to the case that G is compact. It turns out that, if G is not compact, each self-conjugate element T of G gives rise to a self-conjugate element

In this note, we extend our exploration of the essential S-spectrum and introduce the class of Jeribi essential pseudo S-spectrum for bounded right linear operators over the right quaternionic Hilbert space. We explore several properties and the stability of the essential S-spectrum and essential pseudo S-spectrum of this class. Furthermore, we study relationships between the Jeribi essential S-spectrum and other essential S-spectra while investigating the correlation of the essential S-spectrum of this class for 2 × 2 block operator matrices with their components and formulating a characterization of the Jeribi essential S-spectrum for the summation of two bounded linear operators. The study concludes by introducing the notion of essential pseudo S-spectrum of this class and exploration of some of the algebraic properties.

We extend the discussion of projective group representations in quaternionic Hilbert space that was given in our recent book. The associativity condition for quaternionic projective representations is formulated in terms of unitary operators and then analyzed in terms of their generator structure. The multi-centrality and centrality assumptions are also analyzed in generator terms, and implications of this analysis are discussed.

A Representation of Projection Lattices and Their States in Euclidean Space

It is proved that the index of a Fredholm operator between non-Archimedean Fréchet spaces is preserved under compact perturbations. A similar result is shown for Fredholm operators between non-Archimedean polar regular LF-spaces.

Let A be a real B ∗ {B^ \ast } -algebra containing a ∗ \ast -subalgebra that is ∗ \ast -isomorphic to the real quaternion algebra H \mathbb {H} . Suppose the spectrum of every self-adjoint element in A is contained in the real line. Then it is proved that there exists a quaternionic Hilbert space X and an isometric ∗ \ast -isomorphism π \pi of A onto a closed ∗ \ast -subalgebra of B L ( X ) BL(X) , the algebra of all bounded linear operators on X. If, in addition to the above hypotheses, every element in A is normal, then A is also proved to be isometrically ∗ \ast -isomorphic to C ( Y , H ) C(Y,\mathbb {H}) , the algebra of all continuous H \mathbb {H} -valued functions on a compact Hausdorff space Y.

Let (Mz1, ... , MZn ) be an n-tuple of shift operators on the polydisk 12(Zn); we compress it to a variety of subspaces of 12(Zn) that are combinatorially constructed. The main result is a multivariate Fredholm index formula, which links the indices of the n-tuples to their combinatorial data in the definitions of the subspaces. In [11], Taylor, using homological algebraic methods, introduced a natural spectral theory for n-tuples of commuting operators acting on Banach spaces. Fredholm theory and index theory are developed with respect to this spectral theory [2, 3, 8]. While this several variable spectral theory is a natural generalization of the classical single operator theory (when n = 1), the techniques of computing these spectra are so different from or, much more complicated than, that of single operators. The problem of deciding the spectral pictures of various classes of n-tuples of operators is very essential in multivariate operator theory and has deep applications to many areas of analysis, such as function theory on domains in Cn . Many examples have been investigated through many different techniques [4, 5, 7, 9, 10]. In this note, we use the homological method to compute the spectra and Fredholm indices of a new, interesting class of n-tuples of operators. We will give an index formula for these n-tuples in terms of combinatorial data in their definition. An interesting consequence of this is that some combinatorial property of the operator forces the index to be nonzero. To be more precise, we denote by T = (T1, . .. , Tn) an n-tuple of commuting operators on the Hilbert space H and by K(T, z) = {Kq (T, z), 0q } the Koszul complex of T at z E Cn, where Kq (T, z) is the space of q-cochains and aq is the differential map at degree q. We refer the reader to Taylor's original papers [1 1, 12] for a more detailed account of these concepts. Recall that a point z in Cn is said to be in the Taylor spectrum a(T), if the Koszul Received by the editors October 6, 1992. 1991 Mathematics Subject Classification. Primary 47D25. Research supported in part by a grant from NSF, DMS 90-02969. (? 1994 American Mathematical Society 0002-9939/94 $1.00 + $.25 per page