View source for Taylor joint spectrum

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t1300501.png" /> be the [[Exterior algebra|exterior algebra]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t1300502.png" /> generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t1300503.png" />, with identity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t1300504.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t1300505.png" /> is the algebra of forms in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t1300506.png" /> with complex coefficients, subject to the collapsing property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t1300507.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t1300508.png" />). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t1300509.png" /> denote the creation operator, given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005010.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005012.png" />). If one declares <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005013.png" /> to be an orthonormal basis, the exterior algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005014.png" /> becomes a [[Hilbert space|Hilbert space]], admitting an orthogonal decomposition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005015.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005016.png" />. Thus, each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005017.png" /> admits a unique orthogonal decomposition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005018.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005020.png" /> have no <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005021.png" /> contribution. It then readily follows that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005022.png" />. Indeed, each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005023.png" /> is a partial isometry, satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005024.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005025.png" />).

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005026.png" /> be a [[Normed space|normed space]], let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005027.png" /> be a commuting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005028.png" />-tuple of bounded operators on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005029.png" /> and set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005030.png" />. One defines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005031.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005032.png" />. Clearly, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005033.png" />, so <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005034.png" />.

The commuting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005035.png" />-tuple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005036.png" /> is said to be non-singular on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005037.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005038.png" />. The Taylor joint spectrum, or simply the Taylor spectrum, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005039.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005040.png" /> is the set

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005041.png" /></td> </tr></table>

The decomposition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005042.png" /> gives rise to a cochain complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005043.png" />, the so-called [[Koszul complex|Koszul complex]] associated to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005044.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005045.png" />, as follows:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005046.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005047.png" /> denotes the restriction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005048.png" /> to the subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005049.png" />. Thus,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005050.png" /></td> </tr></table>

J.L. Taylor showed in [[#References|[a17]]] that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005051.png" /> is a [[Banach space|Banach space]], then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005052.png" /> is compact, non-empty, and contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005053.png" />, the (joint) algebraic spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005054.png" /> (cf. also [[Spectrum of an operator|Spectrum of an operator]]) with respect to the commutant of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005055.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005056.png" />. Moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005057.png" /> carries an analytic [[Functional calculus|functional calculus]] with values in the double commutant of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005058.png" />, so that, in particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005059.png" /> possesses the projection property.

===Example: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005060.png" />.===
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005061.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005062.png" /> admits the following <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005063.png" />-matrix relative to the direct sum decomposition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005064.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005065.png" /></td> </tr></table>

Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005066.png" />. It follows at once that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005067.png" /> agrees with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005068.png" />, the spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005069.png" />.

===Example: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005070.png" />.===
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005071.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005072.png" /></td> </tr></table>

so <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005073.png" />.

Note that since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005074.png" /> is defined in terms of the actions of the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005075.png" /> on vectors of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005076.png" />, it is intrinsically "spatial" , as opposed to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005077.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005078.png" /> and other algebraic joint spectra. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005079.png" /> contains other well-known spatial spectra, like <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005080.png" /> (the point spectrum), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005081.png" /> (the approximate point spectrum) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005082.png" /> (the defect spectrum). Moreover, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005083.png" /> is a commutative Banach algebra, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005084.png" />, with each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005085.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005086.png" /> denotes the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005087.png" />-tuple of left multiplications by the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005088.png" />s, it is not hard to show that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005089.png" />. As a matter of fact, the same result holds when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005090.png" /> is not commutative, provided all the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005091.png" />s come from the centre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005092.png" />.

===Spectral permanence.===
When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005093.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005094.png" />-algebra, say <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005095.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005096.png" /> [[#References|[a9]]]. This fact, known as spectral permanence for the Taylor spectrum, shows that for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005097.png" />-algebra elements (and also for Hilbert space operators), the non-singularity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005098.png" /> is equivalent to the invertibility of the associated Dirac operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t13005099.png" />.

===Finite-dimensional case.===
When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050100.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050101.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050102.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050103.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050104.png" /> denote the left, right and polynomially convex spectra, respectively. As a matter of fact, in this case the commuting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050105.png" />-tuple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050106.png" /> can be simultaneously triangularized as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050107.png" />, and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050108.png" /></td> </tr></table>

===Case of compact operators.===
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050109.png" /> is a commuting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050110.png" />-tuple of compact operators acting on a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050111.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050112.png" /> is countable, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050113.png" /> as the only accumulation point. Moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050114.png" />.

===Invariant subspaces.===
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050115.png" /> is a Banach space, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050116.png" /> is a closed subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050117.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050118.png" /> is a commuting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050119.png" />-tuple leaving <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050120.png" /> invariant, then the union of any two of the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050121.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050122.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050123.png" /> contains the third [[#References|[a17]]]. This can be seen by looking at the long cohomology sequence associated to the Koszul complex and the canonical short exact sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050124.png" />.

===Additional properties.===
In addition to the above-mentioned properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050125.png" />, the following facts can be found in the survey article [[#References|[a10]]] and the references therein:

i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050126.png" /> gives rise to a compact non-empty subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050127.png" /> of the maximal ideal space of any commutative Banach algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050128.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050129.png" />, in such a way that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050130.png" /> [[#References|[a17]]];

ii) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050131.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050132.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050133.png" /> denotes the Harte spectrum;

iii) the upper semi-continuity of separate parts holds for the Taylor spectrum;

iv) every isolated point in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050134.png" /> is an isolated point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050135.png" /> (and, a fortiori, an isolated point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050136.png" />);

v) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050137.png" />, up to approximate unitary equivalence one can always assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050138.png" /> [[#References|[a5]]];

vi) the functional calculus introduced by Taylor in [[#References|[a18]]] admits a concrete realization in terms of the Bochner–Martinelli kernel (cf. [[Bochner–Martinelli representation formula|Bochner–Martinelli representation formula]]) in case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050139.png" /> acts on a Hilbert space or on a [[C*-algebra|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050140.png" />-algebra]] [[#References|[a20]]];

vii) M. Putinar established in [[#References|[a13]]] the uniqueness of the functional calculus, provided it extends the polynomial calculus.

===Fredholm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050141.png" />-tuples.===
In a way entirely similar to the development of Fredholm theory, one can define the notion of Fredholm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050142.png" />-tuple: a commuting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050143.png" />-tuple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050144.png" /> is said to be Fredholm on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050145.png" /> if the associated Koszul complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050146.png" /> has finite-dimensional cohomology spaces. The Taylor essential spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050147.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050148.png" /> is then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050149.png" /></td> </tr></table>

The Fredholm index of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050150.png" /> is defined as the [[Euler characteristic|Euler characteristic]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050151.png" />. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050152.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050153.png" />. In a Hilbert space, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050154.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050155.png" /> is the coset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050156.png" /> in the Calkin algebra for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050157.png" />.

===Example.===
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050158.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050159.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050160.png" />), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050161.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050162.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050163.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050164.png" />).

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 [[#References|[a4]]], [[#References|[a3]]], [[#References|[a6]]], [[#References|[a7]]], [[#References|[a8]]], [[#References|[a16]]], [[#References|[a15]]], [[#References|[a19]]], and [[#References|[a21]]]; for Toeplitz operators with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050166.png" /> symbols acting on bounded pseudo-convex domains, concrete descriptions appear in [[#References|[a11]]].

===Spectral inclusion.===
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050167.png" /> is a subnormal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050168.png" />-tuple acting on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050169.png" /> with minimal normal extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050170.png" /> acting on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050171.png" /> (cf. also [[Normal operator|Normal operator]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050172.png" /> [[#References|[a14]]].

===Left and right multiplications.===
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050173.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050174.png" /> two commuting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050175.png" />-tuples of operators on a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050176.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050177.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050178.png" /> the associated <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050179.png" />-tuples of left and right multiplication operators [[#References|[a5]]],

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050180.png" /></td> </tr></table>

and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050181.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050182.png" /></td> </tr></table>

During the 1980s and 1990s, Taylor spectral theory has received considerable attention; for further details and information, see [[#References|[a2]]], [[#References|[a11]]], [[#References|[a20]]], [[#References|[a10]]], [[#References|[a1]]]. There is also a parallel "local spectral theory" , described in [[#References|[a11]]], [[#References|[a12]]] and [[#References|[a20]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> E. Albrecht, F.-H. Vasilescu, "Semi-Fredholm complexes" ''Oper. Th. Adv. Appl.'' , '''11''' (1983) pp. 15–39 {{MR|0789629}} {{ZBL|0527.47008}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> C.-G. Ambrozie, F.-H. Vasilescu, "Banach space complexes" , Kluwer Acad. Publ. (1995) {{MR|1357165}} {{ZBL|0837.47009}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> 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 {{MR|584284}} {{ZBL|0436.47021}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> C. Berger, L. Coburn, "Wiener–Hopf operators on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050183.png" />" ''Integral Eq. Oper. Th.'' , '''2''' (1979) pp. 139–173 {{MR|0543881}} {{ZBL|0434.47019}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> R. Curto, L. Fialkow, "The spectral picture of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050184.png" />" ''J. Funct. Anal.'' , '''71''' (1987) pp. 371–392 {{MR|0880986}} {{ZBL|0626.47018}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> R. Curto, P. Muhly, "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050185.png" />-algebras of multiplication operators on Bergman spaces" ''J. Funct. Anal.'' , '''64''' (1985) pp. 315–329 {{MR|0813203}} {{ZBL|0583.46049}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> R. Curto, N. Salinas, "Spectral properties of cyclic subnormal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050186.png" />-tuples" ''Amer. J. Math.'' , '''107''' (1985) pp. 113–138 {{MR|778091}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> R. Curto, K. Yan, "The spectral picture of Reinhardt measures" ''J. Funct. Anal.'' , '''131''' (1995) pp. 279–301 {{MR|1345033}} {{ZBL|0826.47002}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> R. Curto, "Spectral permanence for joint spectra" ''Trans. Amer. Math. Soc.'' , '''270''' (1982) pp. 659–665 {{MR|0645336}} {{ZBL|0508.47039}} {{ZBL|0491.47020}} </TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> 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 {{MR|0976843}} {{ZBL|0827.47005}} </TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> J. Eschmeier, M. Putinar, "Spectral decompositions and analytic sheaves" , ''London Math. Soc. Monographs'' , Oxford Sci. Publ. (1996) {{MR|1420618}} {{ZBL|0855.47013}} </TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top"> K. Laursen, M. Neumann, "Introduction to local spectral theory" , ''London Math. Soc. Monographs'' , Oxford Univ. Press (2000) {{MR|1747914}} {{ZBL|0957.47004}} </TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top"> M. Putinar, "Uniqueness of Taylor's functional calculus" ''Proc. Amer. Math. Soc.'' , '''89''' (1983) pp. 647–650 {{MR|718990}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top"> M. Putinar, "Spectral inclusion for subnormal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050187.png" />-tuples" ''Proc. Amer. Math. Soc.'' , '''90''' (1984) pp. 405–406 {{MR|728357}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top"> N. Salinas, A. Sheu, H. Upmeier, "Toeplitz operators on pseudoconvex domains and foliation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050188.png" />-algebras" ''Ann. of Math.'' , '''130''' (1989) pp. 531–565 {{MR|1025166}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top"> N. Salinas, "The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050189.png" />-formalism and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050190.png" />-algebra of the Bergman <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050191.png" />-tuple" ''J. Oper. Th.'' , '''22''' (1989) pp. 325–343 {{MR|1043731}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top"> J.L. Taylor, "A joint spectrum for several commuting operators" ''J. Funct. Anal.'' , '''6''' (1970) pp. 172–191 {{MR|0268706}} {{ZBL|0233.47024}} </TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top"> J.L. Taylor, "The analytic functional calculus for several commuting operators" ''Acta Math.'' , '''125''' (1970) pp. 1–48 {{MR|0271741}} {{ZBL|0233.47025}} </TD></TR><TR><TD valign="top">[a19]</TD> <TD valign="top"> H. Upmeier, "Toeplitz <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050192.png" />-algebras on bounded symmetric domains" ''Ann. of Math.'' , '''119''' (1984) pp. 549–576 {{MR|744863}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a20]</TD> <TD valign="top"> F.-H. Vasilescu, "Analytic functional calculus and spectral decompositions" , Reidel (1982) {{MR|0690957}} {{ZBL|0495.47013}} </TD></TR><TR><TD valign="top">[a21]</TD> <TD valign="top"> U. Venugopalkrishna, "Fredholm operators associated with strongly pseudoconvex domains in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130050/t130050193.png" />" ''J. Funct. Anal.'' , '''9''' (1972) pp. 349–373 {{MR|0315502}} {{ZBL|0241.47023}} </TD></TR></table>