That means we have that for all alpha in

the A, psi, a alpha equals zero, blah blah blah, equals zero. But this here of course is the

condition for psi to be in the orthogonal complement of the range of A. There this and

this is equivalent to psi is in the range and this is actually also like this is in the range of

A orthogonal because this is zero. Okay so this is very useful. Now another proposition that's very

useful and mind you this has nothing to do with self adjoint operators just to have a densely defined

operator I can construct the adjoint this is valid. Now another proposition which we're going to make

heavy use of and which you really have to memorize actually all the relations from today is if you

have an operator and you have an extension of it so what does this symbol here mean what is the

inclusion symbol between two operators I defined this before and B this is read as B is an extension

of A. What does that mean? It means two things. First of all it means that the domain of B is

bigger than the domain of A and includes it that's the first thing and the second thing is that on

the smaller of the two domains A and B coincide so A alpha is B alpha for all alpha on the smaller

domain because on the bigger domain I can't even compare. So B is an extension of A so essentially

it has to do with B has a bigger domain but where the though in those places where the domain

coincides with the domain of A it is just the operator A. So if you have an extension of A that

is called B then the adjoint operation reverses this inclusion so that means then the adjoint

of A is an extension of the adjoint of B. So let's look at this why this is true proof it's very

simple we'll simply write down so we had just have to check the domains so we simply write down the

definition of the two domains well it's already written over there but for practice this doesn't

harm we'll have to do this several times. The domain of A star is all those sine h such that

there exists an eta in h such that for all alpha in the A it's true that psi A alpha is equal to

eta alpha okay and now for fun let's write down the domain of B star as well and I don't bother

choosing different names for the variables in here you'll see in a second why such that for all well

let's say here for all beta in DB it's true psi A beta no B beta is eta B beta is eta comma beta

right okay so what do we know we know B is an extension of A so the domain of A is a subset

of the domain of B so for these guys here this inclusion holds right now everything behind this

line is a condition on the selection of those elements right now it's a condition such that

for all alpha in the domain this and that holds now this is a bigger domain so which condition is

stronger this for all down here or that condition for all up here what is more difficult to fulfill

the lower one because it's a stronger condition but a stronger condition will kick out more size

is that right done very simple okay so let me frame the important relations so these relations

are very very useful and you should not forget them and these relations here come with virtually

no conditions other than that the A star and the B star are defined which only hinges on A and B

being densely defined okay so this is with virtually no assumptions this is the property

merely of the adjoint question so far

so without these techniques it's almost impossible to even write down the definition of some of the

most heavily used physical observables in quantum mechanics so this is not mathematical

pedantry this key to the understanding we'll have one lecture in which I will at some point in the

future in the near future in which I in which I will construct operators that are important

for quantum physics so that you will see all of this in action but there's no point in providing

examples before we have the full theory there because then otherwise you won't see the forest

for the trees so now the first section was about the adjoint operator now we look at something more

specific we look at the adjoint of so-called symmetric or the adjoints of symmetric operators

well each operator has one adjoint it's clear to say the adjoint of a symmetric operator okay

there's no no confusion here now I first of all need to tell you what a symmetric operator is and

the most beautiful thing about symmetric operators is as soon as I provide you with one in a concrete

case it's very easy to say whether it's symmetric or not okay it will turn out to be very hard to

say with an operator a self adjoint or not although self adjointness is only a subtle

strengthening technically a subtle strengthening of the symmetry condition I'm going to write down