Taylor joint spectrum
Let be the exterior algebra on
generators
, with identity
.
is the algebra of forms in
with complex coefficients, subject to the collapsing property
(
). Let
denote the creation operator, given by
(
,
). If one declares
to be an orthonormal basis, the exterior algebra
becomes a Hilbert space, admitting an orthogonal decomposition
, where
. Thus, each
admits a unique orthogonal decomposition
, where
and
have no
contribution. It then readily follows that
. Indeed, each
is a partial isometry, satisfying
(
).
Let be a normed space, let
be a commuting
-tuple of bounded operators on
and set
. One defines
by
. Clearly,
, so
.
The commuting -tuple
is said to be non-singular on
if
. The Taylor joint spectrum, or simply the Taylor spectrum, of
on
is the set
![]() |
The decomposition gives rise to a cochain complex
, the so-called Koszul complex associated to
on
, as follows:
![]() |
where denotes the restriction of
to the subspace
. Thus,
![]() |
J.L. Taylor showed in [a17] that if is a Banach space, then
is compact, non-empty, and contained in
, the (joint) algebraic spectrum of
(cf. also Spectrum of an operator) with respect to the commutant of
,
. Moreover,
carries an analytic functional calculus with values in the double commutant of
, so that, in particular,
possesses the projection property.
Contents
Example:
.
For ,
admits the following
-matrix relative to the direct sum decomposition
:
![]() |
Then . It follows at once that
agrees with
, the spectrum of
.
Example:
.
For ,
![]() |
so .
Note that since is defined in terms of the actions of the operators
on vectors of
, it is intrinsically "spatial" , as opposed to
,
and other algebraic joint spectra.
contains other well-known spatial spectra, like
(the point spectrum),
(the approximate point spectrum) and
(the defect spectrum). Moreover, if
is a commutative Banach algebra,
, with each
, and
denotes the
-tuple of left multiplications by the
s, it is not hard to show that
. As a matter of fact, the same result holds when
is not commutative, provided all the
s come from the centre of
.
Spectral permanence.
When is a
-algebra, say
, then
[a9]. This fact, known as spectral permanence for the Taylor spectrum, shows that for
-algebra elements (and also for Hilbert space operators), the non-singularity of
is equivalent to the invertibility of the associated Dirac operator
.
Finite-dimensional case.
When ,
![]() |
where ,
and
denote the left, right and polynomially convex spectra, respectively. As a matter of fact, in this case the commuting
-tuple
can be simultaneously triangularized as
, and
![]() |
Case of compact operators.
If is a commuting
-tuple of compact operators acting on a Banach space
, then
is countable, with
as the only accumulation point. Moreover,
.
Invariant subspaces.
If is a Banach space,
is a closed subspace of
and
is a commuting
-tuple leaving
invariant, then the union of any two of the sets
,
and
contains the third [a17]. This can be seen by looking at the long cohomology sequence associated to the Koszul complex and the canonical short exact sequence
.
Additional properties.
In addition to the above-mentioned properties of , the following facts can be found in the survey article [a10] and the references therein:
i) gives rise to a compact non-empty subset
of the maximal ideal space of any commutative Banach algebra
containing
, in such a way that
[a17];
ii) for ,
, where
denotes the Harte spectrum;
iii) the upper semi-continuity of separate parts holds for the Taylor spectrum;
iv) every isolated point in is an isolated point of
(and, a fortiori, an isolated point of
);
v) if , up to approximate unitary equivalence one can always assume that
[a5];
vi) the functional calculus introduced by Taylor in [a18] admits a concrete realization in terms of the Bochner–Martinelli kernel (cf. Bochner–Martinelli representation formula) in case acts on a Hilbert space or on a
-algebra [a20];
vii) M. Putinar established in [a13] the uniqueness of the functional calculus, provided it extends the polynomial calculus.
Fredholm
-tuples.
In a way entirely similar to the development of Fredholm theory, one can define the notion of Fredholm -tuple: a commuting
-tuple
is said to be Fredholm on
if the associated Koszul complex
has finite-dimensional cohomology spaces. The Taylor essential spectrum of
on
is then
![]() |
The Fredholm index of is defined as the Euler characteristic of
. For example, if
,
. In a Hilbert space,
, where
is the coset of
in the Calkin algebra for
.
Example.
If and
(
), then
,
, and
(
).
The Taylor spectral and Fredholm theories of multiplication operators acting on Bergman spaces over Reinhardt domains or bounded pseudo-convex domains, or acting on the Hardy spaces over the Shilov boundary of bounded symmetric domains on several complex variables, have been described in [a4], [a3], [a6], [a7], [a8], [a16], [a15], [a19], and [a21]; for Toeplitz operators with symbols acting on bounded pseudo-convex domains, concrete descriptions appear in [a11].
Spectral inclusion.
If is a subnormal
-tuple acting on
with minimal normal extension
acting on
(cf. also Normal operator),
[a14].
Left and right multiplications.
For and
two commuting
-tuples of operators on a Hilbert space
, and
and
the associated
-tuples of left and right multiplication operators [a5],
![]() |
and
![]() |
![]() |
During the 1980s and 1990s, Taylor spectral theory has received considerable attention; for further details and information, see [a2], [a11], [a20], [a10], [a1]. There is also a parallel "local spectral theory" , described in [a11], [a12] and [a20].
References
[a1] | E. Albrecht, F.-H. Vasilescu, "Semi-Fredholm complexes" Oper. Th. Adv. Appl. , 11 (1983) pp. 15–39 |
[a2] | C.-G. Ambrozie, F.-H. Vasilescu, "Banach space complexes" , Kluwer Acad. Publ. (1995) |
[a3] | C. Berger, L. Coburn, A. Koranyi, "Opérateurs de Wiener–Hopf sur les spheres de Lie" C.R. Acad. Sci. Paris Sér. A , 290 (1980) pp. 989–991 |
[a4] | C. Berger, L. Coburn, "Wiener–Hopf operators on ![]() |
[a5] | R. Curto, L. Fialkow, "The spectral picture of ![]() |
[a6] | R. Curto, P. Muhly, "![]() |
[a7] | R. Curto, N. Salinas, "Spectral properties of cyclic subnormal ![]() |
[a8] | R. Curto, K. Yan, "The spectral picture of Reinhardt measures" J. Funct. Anal. , 131 (1995) pp. 279–301 |
[a9] | R. Curto, "Spectral permanence for joint spectra" Trans. Amer. Math. Soc. , 270 (1982) pp. 659–665 |
[a10] | R. Curto, "Applications of several complex variables to multiparameter spectral theory" J.B. Conway (ed.) B.B. Morrel (ed.) , Surveys of Some Recent Results in Operator Theory II , Pitman Res. Notes in Math. , 192 , Longman Sci. Tech. (1988) pp. 25–90 |
[a11] | J. Eschmeier, M. Putinar, "Spectral decompositions and analytic sheaves" , London Math. Soc. Monographs , Oxford Sci. Publ. (1996) |
[a12] | K. Laursen, M. Neumann, "Introduction to local spectral theory" , London Math. Soc. Monographs , Oxford Univ. Press (2000) |
[a13] | M. Putinar, "Uniqueness of Taylor's functional calculus" Proc. Amer. Math. Soc. , 89 (1983) pp. 647–650 |
[a14] | M. Putinar, "Spectral inclusion for subnormal ![]() |
[a15] | N. Salinas, A. Sheu, H. Upmeier, "Toeplitz operators on pseudoconvex domains and foliation ![]() |
[a16] | N. Salinas, "The ![]() ![]() ![]() |
[a17] | J.L. Taylor, "A joint spectrum for several commuting operators" J. Funct. Anal. , 6 (1970) pp. 172–191 |
[a18] | J.L. Taylor, "The analytic functional calculus for several commuting operators" Acta Math. , 125 (1970) pp. 1–48 |
[a19] | H. Upmeier, "Toeplitz ![]() |
[a20] | F.-H. Vasilescu, "Analytic functional calculus and spectral decompositions" , Reidel (1982) |
[a21] | U. Venugopalkrishna, "Fredholm operators associated with strongly pseudoconvex domains in ![]() |
Taylor joint spectrum. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Taylor_joint_spectrum&oldid=14902