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What is the modern classification of linear operators?

Especially interesting regarding linear operators on numerical functions.

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@Axixx: "this does not work in Hilbert spaces." It works in all vector spaces if you don't further specify what type of operator. Your question says nothing about Hilbert space, or what spaces of functions you're referring to (with what topology for example), or what types of operators. I suspect you're looking for some kind of list like in your question math.stackexchange.com/questions/17284/…, but neither question seems well defined in the present formulation. – Jonas Meyer Jan 28 '11 at 19:24
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@Anixx: I will try to be more clear. If you are concerned with all linear operators on a vector space, then any additional structure the vector space may have (such as consisting of differentiable functions) becomes almost irrelevant. There is no better classification in general than the definition of linearity. There are lots of particular types of linear operators on particular spaces (just like polynomials and your other examples from you comment are some particular types of functions). Are you just asking for a list a bunch of types of linear operators? If not, then what precisely? – Jonas Meyer Jan 28 '11 at 19:33
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I am voting to close as not-a-real-question. – Mariano Suárez-Alvarez♦ Jan 28 '11 at 20:43
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@Anixx: The problem is that you are being very unclear and it is difficult to tell exactly what it is you are asking, or what kind of response you are expecting. Your comments so far, rather than clarifying the situation, have made it all the murkier (at least to me). So, the question right now is "ambiguous, unclear, vague... and impossible to answer in its current form", because you are being ambiguous, unclear, and vague about what it is you are actually trying to ask. – Arturo Magidin Jan 28 '11 at 21:40
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@Anixx, if you asked «What is the modern classification of functions?» it would make as little sense as this... – Mariano Suárez-Alvarez♦ Jan 28 '11 at 21:52
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closed as not a real question by Mariano Suárez-Alvarez♦, Jonas Meyer, Andres Caicedo, Arturo Magidin, Mike Spivey Jan 29 '11 at 0:43

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, see the FAQ.

1 Answer

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There are any number of classes of linear operators that people are interested in, depending on what situation you are in. Some examples:

If your space comes equipped with an inner product, one has:

For normed spaces you have bounded operators. Etc.

Is this more or less what you are thinking of?

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Interesting, but what are some examples of the mentioned operators on the space of, say, continuous functions? Are there specific classifications of linear operators on the functional spaces? – Anixx Jan 28 '11 at 20:02
@Anixx: I'm not a functional analyst, so I confess I don't know. But if this is the kind of thing you were looking for, then it's not really a "classification", but rather a collection of "distinguished classes". As for examples, as the Wikipedia pages points out: momentum, angular momentum, and spin are self-adjoing operators in Quantum Mechanics; the Fourier transform is an example of a unitary operator; orthogonal projections are obtained by considering any proper subspace; and so on. – Arturo Magidin Jan 28 '11 at 20:13
I am interested for examples on functional spaces. By the way how the operators from this question math.stackexchange.com/questions/17284/… correlate with the abovementioned classes? – Anixx Jan 28 '11 at 20:16
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@Anixx. Sigh; clearly, I'm having a lot of trouble figuring out just what it is you are actually asking. Though, judging by the responses you have received so far, it would appear you are having a lot of trouble actually asking what you want to ask. Here, you apparently already knew the answer to the question you ask ("a special name for linear forms containing derivatives of certain order?"), so I wasted my time thinking you were asking a question instead of giving an example. – Arturo Magidin Jan 28 '11 at 21:08
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@Anixx: Setting aside the issue that there is no such thing as " the functional space"... Look: You asked "What is the name of the class of operators such that blah?" We said "Differential operators." You replied "Isn't 'differential operators' a class of operators?" Sorry, but the word "Duh" comes to mind. Can you perhaps see why people are finding this frustrating, why your comments are murkying the waters rather than clarfying them, and why some (among them, now, myself) think it is hard to tell just what it is you are asking? – Arturo Magidin Jan 28 '11 at 22:06
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