# 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.

### 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 " Integral Eq. Oper. Th. , 2 (1979) pp. 139–173 |

[a5] | R. Curto, L. Fialkow, "The spectral picture of " J. Funct. Anal. , 71 (1987) pp. 371–392 |

[a6] | R. Curto, P. Muhly, "-algebras of multiplication operators on Bergman spaces" J. Funct. Anal. , 64 (1985) pp. 315–329 |

[a7] | R. Curto, N. Salinas, "Spectral properties of cyclic subnormal -tuples" Amer. J. Math. , 107 (1985) pp. 113–138 |

[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 -tuples" Proc. Amer. Math. Soc. , 90 (1984) pp. 405–406 |

[a15] | N. Salinas, A. Sheu, H. Upmeier, "Toeplitz operators on pseudoconvex domains and foliation -algebras" Ann. of Math. , 130 (1989) pp. 531–565 |

[a16] | N. Salinas, "The -formalism and the -algebra of the Bergman -tuple" J. Oper. Th. , 22 (1989) pp. 325–343 |

[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 -algebras on bounded symmetric domains" Ann. of Math. , 119 (1984) pp. 549–576 |

[a20] | F.-H. Vasilescu, "Analytic functional calculus and spectral decompositions" , Reidel (1982) |

[a21] | U. Venugopalkrishna, "Fredholm operators associated with strongly pseudoconvex domains in " J. Funct. Anal. , 9 (1972) pp. 349–373 |

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Taylor joint spectrum.

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