
Title: Closable Operators

Series: Unbounded Operators

YouTubeTitle: Unbounded Operators  Part 4  Closable Operators

Bright video: https://youtu.be/UEa31BaKbe4

Dark video: https://youtu.be/JNXZlX0vqGo

PrintPDF: Download printable PDF version

Thumbnail (bright): Download PNG

Timestamps

Subtitle in English
1 00:00:00,286 –> 00:00:04,059 Hello and welcome back to unbounded operators.
2 00:00:04,259 –> 00:00:07,716 The video series in the topic of functional analysis
3 00:00:07,916 –> 00:00:12,987 and in today’s part 4 we will talk about so called closable operators.
4 00:00:13,187 –> 00:00:15,758 However, before we start with the content
5 00:00:15,829 –> 00:00:21,980 I first want to thank all the nice people who support this channel on Steady, here on YouTube or on Patreon
6 00:00:22,180 –> 00:00:28,969 and as a reward for supporting me, you get additional material which you can find with the link in the description.
7 00:00:29,443 –> 00:00:35,130 Ok, then let’s start going into the topic by recalling what a closed operator is.
8 00:00:35,686 –> 00:00:40,392 The common name is just T and we need two normed spaces X and Y
9 00:00:40,592 –> 00:00:45,832 and now it’s important to note that we also have a domain for the operator called D(T).
10 00:00:46,032 –> 00:00:50,986 Indeed, this is a well defined linear subspace in the normed space X
11 00:00:51,486 –> 00:00:56,389 and please don’t forget, for us an operator is always a linear map
12 00:00:56,589 –> 00:01:03,261 and now we speak of a closed operator if the graph of this linear map is a closed set.
13 00:01:03,461 –> 00:01:11,513 More precisely we call the graph G_T and it should be a closed set inside the normed space X times Y.
14 00:01:12,200 –> 00:01:19,100 Ok and now if you know some topology or if you know how to deal with closed sets in the metric space,
15 00:01:19,300 –> 00:01:23,542 then you know each set is a subset of a closed set.
16 00:01:23,971 –> 00:01:31,064 In other words if this set is not closed, we can always replace that with the closure of the set
17 00:01:31,486 –> 00:01:35,992 and this fact leads us to so called closable operators.
18 00:01:36,700 –> 00:01:43,708 So this is a more general term. Meaning that each closed operator is always also a closable operator.
19 00:01:44,357 –> 00:01:50,703 However it’s still a requirement. Not every operator is indeed a closable operator
20 00:01:51,143 –> 00:01:55,049 and of course we immediately see that in the definition.
21 00:01:55,249 –> 00:02:01,568 So what we do is that we take the graph G_T as before and then we form the closure of it
22 00:02:01,768 –> 00:02:06,498 and please note in the case of a closed operator this doesn’t change anything
23 00:02:06,698 –> 00:02:14,006 and now what we want for a closable operator is that this new set here is also the graph of an operator.
24 00:02:14,206 –> 00:02:21,507 This means by doing the closure in the space X times Y, we don’t destroy the structure of a graph
25 00:02:22,029 –> 00:02:28,270 and now in the case that this is a well defined graph, we call the operator T with an overline
26 00:02:28,557 –> 00:02:34,047 and moreover we also call this operator the closure of the operator T
27 00:02:34,247 –> 00:02:39,411 and of course by definition the closure of T is a closed operator.
28 00:02:39,957 –> 00:02:48,271 Hence you can remember an operator is called closable, if the closure of the graph defines an operator as well.
29 00:02:48,471 –> 00:02:53,755 So indeed, this is what we have here as the definition for the notion closable
30 00:02:54,071 –> 00:02:59,370 and in the next step I want to reformulate that with the help of sequences.
31 00:02:59,971 –> 00:03:06,156 In fact this is a similar thing we have already done for closed operators in the last video
32 00:03:06,686 –> 00:03:12,871 and I can already tell you, this is a very helpful description you can use in proofs and examples.
33 00:03:13,357 –> 00:03:18,602 You will see that very soon when we will discuss the first example of a closable operator,
34 00:03:19,329 –> 00:03:25,555 but first let’s see how we can use sequences to describe this property for describing the graph of T.
35 00:03:26,300 –> 00:03:31,572 Therefore let’s think what it means that the closure of T is a graph again.
36 00:03:31,772 –> 00:03:41,388 So for example it’s not possible that the pair (0,0) and the pair (0,y), both lie in this set
37 00:03:41,588 –> 00:03:47,571 and of course what we mean is, it’s not possible for a y that is not 0.
38 00:03:48,200 –> 00:03:55,841 Indeed, you should see this is not possible for a well defined map, because one input gets two different outputs.
39 00:03:56,471 –> 00:04:04,819 Moreover this description here already explains everything that could happen with the closure here, because T is a linear map.
40 00:04:05,200 –> 00:04:14,171 Hence we already know that (0,0) lies in this set here and we could also choose any other point then 0 for the input,
41 00:04:14,229 –> 00:04:20,829 because of the linearity or in other words, if we have this double problem here at any other point,
42 00:04:21,000 –> 00:04:24,345 we can just translate it back to the input 0.
43 00:04:24,545 –> 00:04:30,274 Therefore now we can write that being a graph here is equivalent to the fact:
44 00:04:30,786 –> 00:04:37,353 if (0,y) lies in the closure of this graph, then y has to be 0 as well
45 00:04:37,800 –> 00:04:42,351 and exactly this statement we can now describe with sequences.
46 00:04:42,686 –> 00:04:49,862 So what you need to know is, that all the points in the closure we can approximate with points from G_T.
47 00:04:50,400 –> 00:04:57,190 Therefore we would take any sequence x_n from the domain D(T) that converges to 0
48 00:04:57,914 –> 00:05:06,252 Which already means that we want to approximate this 0 here and moreover we also want to approximate this y.
49 00:05:06,557 –> 00:05:11,509 Hence Tx_n should converge to a fixed y in Y.
50 00:05:12,029 –> 00:05:19,975 Ok, so this is the premise. We approximate this point from the inside and now the conclusion is that y has to be 0.
51 00:05:20,175 –> 00:05:25,157 So this is exactly the same thing as before just written with sequences.
52 00:05:25,614 –> 00:05:33,543 Ok and this is what you should remember. A closable operator is exactly characterized with this sequence property here.
53 00:05:34,143 –> 00:05:38,649 Indeed, how you can check that, I will show you in an example soon.
54 00:05:38,986 –> 00:05:45,214 However before we do that I first want to show you how we can define the closure as an operator now
55 00:05:45,900 –> 00:05:50,426 and of course this now only works for a closable operator T
56 00:05:50,626 –> 00:05:57,616 and for this definition we need 2 ingredients. First the domain and then the actual definition of Tbar.
57 00:05:58,457 –> 00:06:03,490 Ok and now the domain consists all the points x in X that fulfill:
58 00:06:03,690 –> 00:06:08,219 that first we can approximate them with points from D(T).
59 00:06:08,600 –> 00:06:17,720 This means the sequence x_n here converges to the point x and in addition we also want that the images converge
60 00:06:18,329 –> 00:06:25,242 and now we know by the property above that this limit here does not depend on which sequence we choose.
61 00:06:25,600 –> 00:06:31,003 More precisely this limit here is uniquely given by the chosen x.
62 00:06:31,486 –> 00:06:37,680 Hence this already explains how we can define the value Tbar_x.
63 00:06:38,314 –> 00:06:42,330 So we just say, this is given by the limit from the righthand side
64 00:06:42,900 –> 00:06:48,090 and please don’t forget, this is only possible because T was a closable operator.
65 00:06:48,471 –> 00:06:54,748 Ok, so now what you see here is, we have a well defined operator called the closure of T
66 00:06:54,948 –> 00:06:59,887 and moreover, you should see it’s an extension of the original operator T
67 00:07:00,429 –> 00:07:05,548 and you already know, the usual notation for that is with this subset notation.
68 00:07:06,157 –> 00:07:12,032 This is not hard to see, because we just extend the domain here and increase the definition.
69 00:07:12,457 –> 00:07:20,995 However the result here is that this Tbar is indeed the smallest possible extension with a closed operator.
70 00:07:21,429 –> 00:07:27,478 So you could say this is another possible definition for the closure of a closable operator.
71 00:07:27,971 –> 00:07:33,671 Ok, so now we know a lot about closable operators and we can look at an example.
72 00:07:33,886 –> 00:07:37,775 However, I would say let’s put that into the next video.
73 00:07:37,975 –> 00:07:41,848 So I really hope we meet there again and have a nice day. Byebye