1 00:00:00,357 –> 00:00:05,460 Hello and welcome to the video series about unbounded operators.

2 00:00:05,843 –> 00:00:13,261 This is a video course in the field of functional analysis and I would say it’s already very special and advanced.

3 00:00:13,461 –> 00:00:21,984 Therefore in order to understand this course in the best way, you should already have good knowledge in linear algebra and real analysis

4 00:00:22,486 –> 00:00:27,933 and of course you can always refresh your memory by watching my videos about these topics

5 00:00:28,133 –> 00:00:35,048 and moreover if you watched my functional analysis course, you already learned a lot about bounded operators.

6 00:00:35,514 –> 00:00:42,747 This means this course here expands the functional analysis course by so called unbounded operators.

7 00:00:43,157 –> 00:00:50,418 Now indeed it turns out that these operators are not easy to describe, but they occur a lot in applications.

8 00:00:50,957 –> 00:00:53,652 However, before I start with the motivation here,

9 00:00:53,852 –> 00:01:00,070 first I really want to thank all the nice people who support this channel on Steady, that made this video course possible.

10 00:01:00,600 –> 00:01:05,610 Since this is such a special topic, which will not generate many views on YouTube.

11 00:01:05,743 –> 00:01:09,986 I can only do it, because of your support on Steady or Patreon

12 00:01:10,186 –> 00:01:16,196 and as a thank you, you will always find the PDF versions and quizzes for the videos on my Webpage.

13 00:01:17,014 –> 00:01:23,368 Ok, then let’s start with the topic and let’s clear up why unbounded operators are so important.

14 00:01:23,771 –> 00:01:26,437 In fact we immediately have a mathematical reason,

15 00:01:26,637 –> 00:01:32,771 because when we deal with partial differential equations we immediately see unbounded operators.

16 00:01:32,971 –> 00:01:41,267 So the whole theory about partial differential equations is much easier to understand when you know what unbounded operators are

17 00:01:41,757 –> 00:01:45,846 and obviously we will explain these connections in this course.

18 00:01:46,214 –> 00:01:50,943 Now, historically the next one might be the biggest motivation after all,

19 00:01:51,071 –> 00:01:55,619 because you need unbounded operators to describe quantum mechanics.

20 00:01:56,100 –> 00:02:00,908 So you might already know that in this physical theory you need operators.

21 00:02:00,971 –> 00:02:07,756 So you definitely need functional analysis, but it turns out that you can’t avoid unbounded operators.

22 00:02:07,956 –> 00:02:14,888 Indeed, one needs 2 special objects as operators and we call them X and P.

23 00:02:15,088 –> 00:02:23,761 So for example for one particle the one measures the position of the particle and the other operator measures the momentum of the particle

24 00:02:24,157 –> 00:02:30,412 and now the physical reality tells us that the order of these two measurements matters.

25 00:02:30,612 –> 00:02:34,509 This means XP is different from PX

26 00:02:34,943 –> 00:02:41,477 and of course by using functional analysis we can describe this as a composition of operators.

27 00:02:41,971 –> 00:02:49,639 However now the order of the composition matters in the sense that the difference is proportional to the identity operator.

28 00:02:50,200 –> 00:02:56,097 More precisely we also have the imaginary unit here, but this is not the important part now.

29 00:02:56,297 –> 00:03:02,717 The crucial thing here is that we can not satisfy this equation with just bounded operators.

30 00:03:03,071 –> 00:03:09,015 This means we have to extend the notion of a linear operator to do quantum mechanics

31 00:03:09,215 –> 00:03:13,787 and in fact we will see that with unbounded operators this works.

32 00:03:14,243 –> 00:03:21,151 However together with the unbounded operators we get other technical difficulties into the theory

33 00:03:21,351 –> 00:03:25,627 and we will immediately see this with the first definition here,

34 00:03:25,827 –> 00:03:29,786 but let’s first start with some objects we should already know.

35 00:03:29,986 –> 00:03:35,673 Namely the first things we need are two normed spaces we call X and Y.

36 00:03:36,086 –> 00:03:42,150 So we have 2 vector spaces over the same field and they also carry a norm

37 00:03:42,350 –> 00:03:46,846 and you might know we either have real vector spaces or complex ones.

38 00:03:47,046 –> 00:03:52,778 Therefore if we call the field F it’s either the real number line or the complex plane

39 00:03:52,978 –> 00:03:58,357 and I can already tell you, most of the time you can just thing of complex vector spaces here.

40 00:03:58,929 –> 00:04:06,408 Ok, but now immediately something new arises. Namely we need a domain for our operator

41 00:04:06,608 –> 00:04:11,269 and this will be a subspace in X and we call it D.

42 00:04:11,469 –> 00:04:16,130 Hence, now the map we consider and call T is a linear one

43 00:04:16,330 –> 00:04:21,886 and you can put in elements from the subspace D and you get out elements from Y

44 00:04:22,227 –> 00:04:27,856 and now in this course we keep it simple. Such a linear map, we always call an operator.

45 00:04:28,343 –> 00:04:35,352 So please keep that in mind. If we talk about unbounded operators, we implicitly mean linear operators.

46 00:04:35,552 –> 00:04:42,200 Ok and now it’s a good time to also talk about other notations people use to denote such an operator.

47 00:04:42,729 –> 00:04:50,392 Now, the first one is very similar. Some people just use the subset relation here, in the definition of the operator again.

48 00:04:50,592 –> 00:04:55,957 This makes sense, because often the space X is the important normed space here.

49 00:04:56,157 –> 00:05:02,621 However, some other people abuse this notation a little bit by omitting the actual domain here

50 00:05:02,821 –> 00:05:08,015 and if they are nice, they at least say somewhere, that this T has the domain D.

51 00:05:08,215 –> 00:05:14,285 So you can recognize with this that some people see an operator as a pair of 2 things.

52 00:05:14,485 –> 00:05:20,738 First they say they have an action defined by T and then a restriction to a domain D.

53 00:05:20,938 –> 00:05:25,491 So this keeps it very short, but you also often see that in some books.

54 00:05:26,029 –> 00:05:30,589 Or similarly instead of a pair, one writes T with domain D,

55 00:05:30,789 –> 00:05:35,882 but instead of the word domain, one also often uses D(T).

56 00:05:36,200 –> 00:05:43,629 Ok, maybe that’s good enough for the different notations you could stumble upon, but let’s go back to the actual definition here.

57 00:05:44,471 –> 00:05:51,240 Now for this course a very important term will be given by so called densely defined operators

58 00:05:51,800 –> 00:05:57,132 and you might already expect that these will be the operators we will discuss in the next videos

59 00:05:57,786 –> 00:06:04,462 and the definition should be very clear. T is called densely defined, if domain D is dense.

60 00:06:04,900 –> 00:06:13,210 More precisely this means if you take the closure in the space X, then you get the full space X out.

61 00:06:13,686 –> 00:06:20,218 So maybe the domain of the operator is not the whole space, but it’s very close to that in this sense

62 00:06:20,629 –> 00:06:28,623 and here please note, this thing here is only interesting in infinite dimensions, because D is still a linear subspace.

63 00:06:28,823 –> 00:06:35,957 So only in infinite dimensions, we could have smaller subspaces that are still dense in the whole space.

64 00:06:36,671 –> 00:06:42,424 Ok by knowing that we can also write down 2 important subspaces we use all the time

65 00:06:42,624 –> 00:06:47,041 and of course I mean range and kernel for the operator T.

66 00:06:47,471 –> 00:06:54,457 So the explicit definitions you might already know, but because we have to consider a domain here, I want to write them down.

67 00:06:54,657 –> 00:07:00,057 Now first the range of T is given by all the elements we hit on the right-hand side.

68 00:07:00,257 –> 00:07:04,674 This means we have to go through all lower case x in D

69 00:07:04,874 –> 00:07:08,068 and what we get is a subspace in Y.

70 00:07:08,268 –> 00:07:12,086 So please remember that. It’s still a linear subspace.

71 00:07:12,457 –> 00:07:17,830 Ok and now for the kernel we have to take all the points x that are sent to 0.

72 00:07:18,300 –> 00:07:22,566 Therefore what we get here is a linear subspace in X.

73 00:07:22,766 –> 00:07:28,504 Ok and with that we have it. We have the range of the operator T and the kernel of T

74 00:07:28,704 –> 00:07:33,838 and now in the next step we can finally recall the definition of boundedness

75 00:07:34,314 –> 00:07:39,917 and then of course if you know what bounded means, you also know what unbounded means.

76 00:07:40,117 –> 00:07:48,863 So now bounded means we find a constant c. Which we could choose greater than 0, but it should be a finite number

77 00:07:49,063 –> 00:07:55,909 and now for every point x in the domain we can calculate Tx in the norm of Y

78 00:07:56,109 –> 00:08:00,620 and now this one should always be bounded by the constant c.

79 00:08:00,820 –> 00:08:06,115 More precisely it’s c times the norm of x in the space X

80 00:08:06,871 –> 00:08:13,278 and here you should recognize if you calculate the operator norm you have learned in the functional analysis course,

81 00:08:13,478 –> 00:08:17,857 you see this operator norm is bounded by the constant c

82 00:08:18,286 –> 00:08:22,251 and therefore the operator T is called bounded

83 00:08:22,671 –> 00:08:27,675 and now if the operator norm is infinite, we call the operator unbounded

84 00:08:28,057 –> 00:08:30,892 or more precisely we can do the negation here.

85 00:08:30,914 –> 00:08:41,689 Which means for all constants we find at least one point x such that Tx in the norm is greater than c times x in the norm.

86 00:08:41,943 –> 00:08:47,274 This means the ratio Tx to x exceeds every bound

87 00:08:47,886 –> 00:08:53,067 and therefore the operator norm as a number would give infinity in this case.

88 00:08:53,267 –> 00:08:59,312 Indeed this can definitely happen and we will see some important examples in the next video.

89 00:08:59,512 –> 00:09:05,797 However here I first want to recall an important equivalence from the functional analysis course.

90 00:09:05,997 –> 00:09:13,756 Namely a linear map is bounded if and only if it’s continuous at all the points in the domain.

91 00:09:14,186 –> 00:09:20,777 So you see for operators the terms boundedness and continuity are connected in this way.

92 00:09:21,186 –> 00:09:28,255 Therefore unbounded operators as we want to consider them in this course are not continuous at all.

93 00:09:28,643 –> 00:09:33,915 Indeed, an unbounded operator has to be discontinuous at all points.

94 00:09:34,115 –> 00:09:39,758 So this is important to remember. Unbounded means not continuous at all

95 00:09:39,958 –> 00:09:46,019 and there we see again, This is definitely something that can only happen in infinite dimensions.

96 00:09:46,357 –> 00:09:52,323 Of course in finite dimensions every linear map has to be a continuous map.

97 00:09:52,643 –> 00:09:59,890 However in infinite dimensions we have so many directions that the continuity property can fail

98 00:10:00,090 –> 00:10:06,773 and at this point we already know we can not ignore such operators, because they occur in a lot of applications

99 00:10:06,973 –> 00:10:13,120 and therefore I would say let’s look at some examples in the next video and let’s discuss it more.

100 00:10:13,320 –> 00:10:16,100 So have a nice day and bye bye!

Q1: Let $X$ be a normed space and $U \subseteq X$ be a subspace. What is correct for a linear map $T: U \rightarrow X$ that we call an operator on $X$.

A1: $T$ has domain $U$

A2: $T$ is bounded

A3: $T$ is densely defined

A4: $T$ is unbounded

Q2: Let $X$ be a normed space and $U \subseteq X$ be a dense subspace. Let $T: U \rightarrow X$ be an operator that is continuous at the origin. What is not correct?

A1: $T$ is unbounded.

A2: $T$ is bounded.

A3: $T$ is densely defined.

A4: $T$ is continuous at every point in $U$.

Q3: Let $X$ be a finite-dimensional normed space and $U \subseteq X$ be a subspace. Let $T: U \rightarrow X$ be an operator. Can $T$ be unbounded?

A1: No, every linear map is continuous in this case.

A2: Yes, this is possible.

A3: Yes, if $T$ is the zero operator.

A4: One needs more information.