Talk:Holomorphic functional calculus

Typo -- graphy

The text will be easier to read if the typography is changed from $T$ to T. Bo Jacoby 21:33, 29 January 2006 (UTC)

Be my guest. :) As long as you don't introduce extra PNG images, as per the math style manual. Oleg Alexandrov (talk) 03:26, 30 January 2006 (UTC)

example in article

the example re logarithms is maybe not the best e.g. to motivate the construction. it is known from matrix theory that every nonsingular matrix has a logarithm. and the holomorphic functional calculus naturally doesn't extend this result. the example and the accompanying remarks could lead a reader to believe otherwise. if the point is one can not simply replace the scalar variable by the operator T in a series, well, one can, as long as the series converges in a sufficiently large neighborhood containing σ(T), in which case it coincides with the functional calculus(as one would hope). Mct mht 00:54, 1 August 2006 (UTC)

I agree. To deal with the logarithm in the scalar case, it's common to use both power series and the Cauchy formula. What's the point of distinguishing between them in the operator case? Is one better than the other and it's motivated by the logarithm? What is different or harder about a finite radius of convergence in the operator case that is not already present in the scalar case?

In other words, the suggested motivation doesn't seem to have anything to do with the special difficulties of operators, but just with a known limitation of power series in complex analysis. 178.39.122.125 (talk) 19:22, 4 January 2017 (UTC)

Invariant subspace decomposition

Very good page! Only one question: the functions e_i used in the proof of the invariant subspace decomposition can hardly be called holomorphic. How do we know that we can use the homomorphism property? —Preceding unsigned comment added by Octonion (talk • contribs) 10:50, 28 July 2009 (UTC)

Uncorrect statement

It is said in the article that :

"It is desirable to have a functional calculus that allows one to define, for a non-singular T,

   \ln (T)\,

such that it coincides with S. This can not be done via power series. For example, the logarithmic series

   \ln(z+1) = z - \frac{z^2}{2} + \frac{z^3}{3} - \cdots 

converges only on the open unit disk. Substituting T for z in the series fails to give a well-defined ln (T + I) for any invertible T + I with

   \| T \| \geq 1.\,

Thus a more general functional calculus is needed."

Even though this is a nice way of motivating the following sections of the article, the above statements are not fully correct in that it is easy and simple to modify the power-series of \ln(T+I) so that it converges for any T matrix, thus a more general functional calculus is NOT really needed for this purpose. I am not denying the usefulness of holomorphic functional calculus but I think we should modify this motivation section to make it really motivating.... If nobody is against this modification I will carry it out. —Preceding unsigned comment added by 163.1.246.64 (talk) 16:21, 28 February 2011 (UTC)

Two sections are very redundant

The sections

4.2 Spectral projections

4.3 Invariant subspace decomposition

seem to re-derive the same material with a lot of overlap. They should be merged and tightened! 178.39.122.125 (talk) 19:28, 4 January 2017 (UTC)

Removed statements

I removed this nice text from the beginning of the section "Spectral Considerations" because it didn't seem relevant to the section. (It already represents a slight modification of the original introduction.)

The section "Spectral Considerations needs a little introduction, but I can't think what to write.

The above discussion demonstrates the close relationship between the holomorphic functional calculus of an operator T and its spectrum σ(T). These relationships are true for all bounded operators. In the more restrictive class of bounded normal operators, the well-known spectral theorem can be given an alternative, equivalent formulation as a functional calculus where the functions only have to be continuous."

178.39.122.125 (talk) 20:07, 4 January 2017 (UTC)

Unclear statement

The section Functional calculus for a bounded operator contains this passage:

"The full definition of the functional calculus is as follows: For T ∈ L(X), define

$f(T)={\frac {1}{2\pi i}}\int \nolimits _{\Gamma }{\frac {f(\zeta )}{\zeta -T}}\,d\zeta ,$

where f is a holomorphic function defined on an open set D ⊂ C which contains σ(T), and Γ = {γ₁, ..., γm} is a collection of disjoint Jordan curves in D bounding an "inside" set U, such that σ(T) lies in U, and each γi is oriented in the boundary sense."

It is difficult enough to convey advanced mathematical concepts using English.

Please do not use non-words like "inside" — a word with quotation marks around it that is obviously intended to mean something like inside but somehow different from inside at the same time in a way that no reader can guess.

Just stick to actual words. If this means that more words are needed to explain something, so be it — at least the explanation will be clear to readers. 2601:200:C000:1A0:25A0:5BDF:59DB:4E96 (talk) 23:37, 15 September 2022 (UTC)