1 00:00:00,314 –> 00:00:03,951 Hello and welcome back to unbounded operators.

2 00:00:04,151 –> 00:00:08,535 The video series, where we tackle some more topics of functional analysis

3 00:00:08,735 –> 00:00:14,232 and now in today’s part 2 we will look at some examples for unbounded operators.

4 00:00:14,432 –> 00:00:22,914 However, as always before we start I first want to thank all the nice people who support this channel on Steady, here on YouTube or on Patreon.

5 00:00:23,186 –> 00:00:28,038 Ok, then let’s start by recalling what we mean by an operator.

6 00:00:28,386 –> 00:00:33,909 So you should know, it’s just the linear map between 2 normed spaces, X and Y.

7 00:00:34,109 –> 00:00:40,194 Moreover, we also say it has a domain D(T) given by a subspace D.

8 00:00:40,394 –> 00:00:45,321 This actually means that we only have a map from D into Y

9 00:00:45,714 –> 00:00:49,506 and please don’t forget it’s always a linear map.

10 00:00:49,706 –> 00:00:55,445 Ok and now such operators here could be either bounded or unbounded

11 00:00:55,486 –> 00:00:58,490 and this we can measure wit the operator norm.

12 00:00:58,690 –> 00:01:02,962 So this we already know. So let’s state some other important fact.

13 00:01:03,386 –> 00:01:09,002 Namely we can consider the case that the kernel of our operator is trivial.

14 00:01:09,371 –> 00:01:14,642 This means it’s equal to the smallest possible subspace, which is the zero vector.

15 00:01:14,842 –> 00:01:19,580 In particular this means that the linear map T is injective

16 00:01:19,986 –> 00:01:26,363 and from that we now can conclude that we can write down an inverse map T to the power -1.

17 00:01:26,563 –> 00:01:31,388 Now of course here we change the direction. So now we map Y into X

18 00:01:31,588 –> 00:01:36,379 and we can do that because we can write down a suitable domain

19 00:01:36,579 –> 00:01:44,223 and indeed in general the domain is not equal to the whole space Y, but only equal to the range of T.

20 00:01:44,423 –> 00:01:51,906 This makes sense because with this restriction here on the right-hand side we would get the surjectivity for our linear map.

21 00:01:52,106 –> 00:02:01,549 So in short what we have learned here is if the kernel of T is trivial, the inverse T^-1 is always defined as an operator.

22 00:02:01,749 –> 00:02:10,165 However please note this operator here can be unbounded even in the case that the original operator T was bounded.

23 00:02:10,365 –> 00:02:16,334 Nevertheless you see this makes our lives a little bit simpler for defining operators

24 00:02:16,843 –> 00:02:20,941 and with that I would say we are ready for discussing examples

25 00:02:21,141 –> 00:02:28,047 and for the start I want to keep it simple. So let’s say that X and Y represent the same normed space

26 00:02:28,247 –> 00:02:31,972 and this one should be the space of continuous functions

27 00:02:32,286 –> 00:02:38,728 and maybe let’s say the continuous functions are only defined on the unit interval 0 to 1

28 00:02:38,928 –> 00:02:46,011 and you might know, the common norm one chooses for this space is the uniform norm, also called the supremum norm

29 00:02:46,211 –> 00:02:50,494 and one just uses an infinity symbol to denote this norm.

30 00:02:50,694 –> 00:02:54,902 Ok, then let’s define an operator that maps X to Y.

31 00:02:55,102 –> 00:03:03,946 So as before we write T from X to Y, but what we actually mean is that T has a domain as a subspace in X

32 00:03:04,146 –> 00:03:12,409 and for this case here, I want to choose the domain as the continuous functions that are also continuously differentiable.

33 00:03:12,609 –> 00:03:17,785 So in short I just want to write C^1 as a subspace in C.

34 00:03:18,100 –> 00:03:27,006 Because then I can simply write that T applied to an element from the domain is defined as the derivative

35 00:03:27,206 –> 00:03:32,508 and if we call the function x we have Tx is equal to x prime.

36 00:03:32,708 –> 00:03:40,344 Now, obviously this is a well defined map and by the properties of the derivative it’s also a linear map.

37 00:03:40,544 –> 00:03:46,342 In other words it’s an operator and now we can show, it’s an unbounded one

38 00:03:46,542 –> 00:03:51,035 and this is not hard at all, because we can just look at some pictures.

39 00:03:51,235 –> 00:03:57,566 So maybe imagine that we have some nice smooth functions here, where the maximum is at 1.

40 00:03:57,766 –> 00:04:02,705 Now, this fact helps for calculating the supremum norm of T.

41 00:04:03,014 –> 00:04:07,729 So I think it’s very helpful to write down the definition of the supremum norm again.

42 00:04:08,271 –> 00:04:15,157 So one possibility is to take the supremum over all possible inputs that have norm 1

43 00:04:15,359 –> 00:04:21,005 and please note all these possible x elements have to come from the chosen domain here

44 00:04:21,205 –> 00:04:25,869 and then we can just take the supremum of the norm Tx.

45 00:04:26,371 –> 00:04:33,385 Ok so this is the general definition of the supremum norm if we just filled in the chosen norm of the two spaces

46 00:04:34,071 –> 00:04:38,294 and then in the next step here we can use the definition of our T.

47 00:04:38,494 –> 00:04:46,358 So you see what goes in here is now the maximal value of the absolute value of the derivative of the function x.

48 00:04:46,857 –> 00:04:54,823 Hence by going back to our picture here, we see that we can define a new function with a higher maximal derivative there.

49 00:04:55,143 –> 00:05:00,114 So you should see the slope here is simply higher than that from the function before

50 00:05:00,457 –> 00:05:06,611 and then you should see we can do that again and again to increase the slope as much as we want

51 00:05:07,114 –> 00:05:13,247 and the important part by doing that is that we don’t increase the supremum norm of the function x.

52 00:05:13,743 –> 00:05:20,180 Ok and then we have to conclude that the supremum of the derivative here is not bounded.

53 00:05:20,380 –> 00:05:25,064 In other words the operator norm of T is equal to infinity.

54 00:05:25,264 –> 00:05:29,093 So indeed we have an unbounded operator here

55 00:05:29,293 –> 00:05:33,583 and you should note this was not a complicated operator at all.

56 00:05:33,929 –> 00:05:40,409 Ok then I would say let’s use this example to define another unbounded operator

57 00:05:40,609 –> 00:05:45,964 and let’s use this same space as X and Y and let’s call the operator S.

58 00:05:46,164 –> 00:05:53,578 Indeed we can also use the same definition as before with the derivative, but now I want to change the domain.

59 00:05:53,778 –> 00:06:00,327 Now, I only want to use some particular continuously differentiable functions for the domain.

60 00:06:00,527 –> 00:06:08,019 So we could write: let’s choose functions x in C^1 that are fixed on the left boundary.

61 00:06:08,219 –> 00:06:12,675 Namely x(0) = 0.

62 00:06:12,875 –> 00:06:20,250 This means now we have a new subspace of continuous functions which is definitely smaller than the one before.

63 00:06:20,971 –> 00:06:26,478 Otherwise you see everything is the same as we had it for the operator T.

64 00:06:26,678 –> 00:06:31,100 Therefore there are some special notations one uses usually for this fact

65 00:06:31,229 –> 00:06:36,227 and this is a little bit strange, because we use an inclusion symbol for operators,

66 00:06:36,929 –> 00:06:41,746 but it makes literally sense if you put in the corresponding domains.

67 00:06:42,229 –> 00:06:51,669 Hence the subset notation here actually means that S in T are essentially the same operator, but S has a smaller domain

68 00:06:51,869 –> 00:06:57,029 and for this reason one says that T is an extension of S.

69 00:06:57,229 –> 00:07:01,375 So this is definitely something you should remember in this context

70 00:07:01,643 –> 00:07:07,135 and on the other hand the operator S is called a restriction for the operator T.

71 00:07:07,443 –> 00:07:13,507 So you see the names make sense and they are always used if we talk about domains.

72 00:07:13,707 –> 00:07:20,067 Indeed, we will see that changing the domain, making it smaller or larger, can change a lot.

73 00:07:20,571 –> 00:07:25,154 For example in this case here the inverse operator changes.

74 00:07:25,354 –> 00:07:30,744 In fact the operator T does not have an inverse, because it’s not injective.

75 00:07:30,944 –> 00:07:35,675 Simply because any constant function has derivative 0.

76 00:07:35,875 –> 00:07:42,347 However this doesn’t happen to S, because the only constant function is the 0 function.

77 00:07:42,886 –> 00:07:46,559 Hence we conclude that S is an injective operator

78 00:07:46,759 –> 00:07:52,101 and as explained before we can write down S^-1.

79 00:07:52,301 –> 00:07:56,865 So this is a well defined operator in the sense of our definition.

80 00:07:57,065 –> 00:08:01,973 Ok, but I can tell you this nice fact here comes with a price.

81 00:08:02,173 –> 00:08:07,936 Namely the domain of the restriction S is much smaller than the original domain.

82 00:08:08,136 –> 00:08:14,602 More precisely the continuously differentiable functions lie dense in the continuous functions

83 00:08:14,957 –> 00:08:22,899 or more precisely the closer of C^1 with respect to the supremum norm is equal to C.

84 00:08:23,099 –> 00:08:27,591 Indeed this is a very nice fact from the continuously differentiable functions.

85 00:08:27,971 –> 00:08:33,423 However with the restriction here on the left-hand side this doesn’t hold anymore.

86 00:08:33,623 –> 00:08:39,537 We immediately see that we can’t approximate any constant function that is not the 0 function

87 00:08:40,114 –> 00:08:44,306 and therefore the operator S is not densely defined

88 00:08:44,506 –> 00:08:50,525 and you might already expect that in most cases we want to have a densely defined operator,

89 00:08:50,725 –> 00:08:55,458 because otherwise we are just too far away from the original space.

90 00:08:55,658 –> 00:09:00,500 Of course in general it would be the best thing to have the whole space as a domain,

91 00:09:00,586 –> 00:09:04,829 but we have already seen this is not possible for a lot of operators

92 00:09:05,043 –> 00:09:12,663 and in fact for most unbounded operators we are interested in it’s not possible to define them on the whole space.

93 00:09:12,863 –> 00:09:16,896 So we always have to weaken that to densely defined operators,

94 00:09:17,614 –> 00:09:21,658 but I would say let’s see how this works out with the next videos.

95 00:09:21,858 –> 00:09:26,100 So I really hope that I see you there and have a nice day. Bye-bye

Q1: Let $X$ be a normed space and $T: X \rightarrow X$ given by $Tx = 1$ for all $x$. What is correct?

A1: $T$ is not an operator.

A2: $T$ is a bounded operator.

A3: $T$ is an unbounded operator.

A4: $T$ is not well-defined.

Q2: Let $X = C([0,1])$ and $T: X \rightarrow X$ an operator with $D(T) = C^1([0,1])$ and $T x = x^\prime + x$. What is correct?

A1: $T$ is unbounded.

A2: $T$ is bounded.

A3: $T$ is not densely defined.

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