Variational formulationa of self adjoint operators can be done and Ritz method can be applied to find solutons of equations having the property.

Unitary operators possess orthogonality and isometry properties and are very helpful

to obtain solutions.

Normal operators have the property that A^* A = A A^*. They include both unitary and self-adjoint operators. Normal operators are unitarily equivalent to diagonal operators, at least over separable Hilbert spaces.

@ David: I know that, whar I am asking is some other peculiar property in the following sense: Take the collection of all self adjoint operators. what is the property of this collection? Just like that

In the operator theory one works very often with self-adjoint linear relations (which are, in a sense, multi-valued self-adjoint operators). These are natural from the geometric point of view, as they are just maximal subspaces which are self-orthogonal wrt to a certain symplectic form, and they are in one-to-one relation with the unitary operators (using the Cayley transform).

The collection of self-adjoint operators is a just a subsollection of the set of self-adjoint linear relations with the propertt that they can be projected one-to-one to a given subspace.

Given a Hilbert space H, the collection of all bounded selfadjoint operators in H makes a linear space over R, but not over C. Probably the most interesting and useful property is that saying that the linear space of bounded selfadjoint operators in H is (real) linearly generated by the cone of positive operators: this is coming from the fact that any bounded selfadjoint operator can be decomposed as the difference of two positive operators (an analog of Jordan-Haar decomposition for measures).

The class of self-adjoint operators have many properties. Among those are: They are compose of an operator and its adjoint. And, their spectrums are all real.

The topological space of Unitary operators on an inifinite dimensional Hilbert space (considered as a closed set of the space of all operators in the norm topology) is contractible. This is a famous theorem of Kuipers and important in topological K-theory.

Note that in finite dimensions the unitary operators are not at all contractible !

See https://en.wikipedia.org/wiki/Kuiper's_theorem

Remark: the space

U(\infty) = \colim U(n) = "\union"_n U(n)

where U(n-1) \subset U(n) by adding a diagonal 1 is not contractible either. This space is often used by topologists.

The normal operators are NOT a vector space. This is because the sum of two normal operators does not have to be normal!

The collection of all bounded self adjoint operators forms an ordered (real) linear space, with respect to the usual order relation on this linear space. One of the basic properties of this ordered vector space is: if U_n is a monotone non decreasing bounded from above sequence in this space, then it admits a least upper bound

U = sup U_n, and U is the pointwise limit of the sequence U_n. Notice that U is also an element of this space. This property has various applications. To mention one of them, I recall the proof of the existence of a unique positive square root of a positive bounded self adjoint operator, which is using the above mentioned property of monotone sequences. On the other hand, there are interesting subspaces of this ordered vector space: for each bounded self adjoint operator one defines the associated bicommutant. This a commutative Banach algebra of self adjoint operators, and an order complete Banach lattice. This last property allows working Hahn-Banach type results for operators having the bicommutant as a target space.

The semi-order on the space of bounded SA linear operators is quite poor. Namely, a closed sub-algebra of the latter is a lattice as far as it is commutative (for details see e.g. Order Properties of Bounded Self-Adjoint Operators by Richard V. Kadison in Proc. AMS vol 2 no 3 (1951), DOI: 10.2307/2031784). However, it has nice topological properties. For instance, it is closed in the weak operator topology. Moreover, the same is true for unbounded SA linear operators, with the correspondent resolvent convergence.

The properties of bicommutant associated to a SA operator of being a commutative algebra and an order-complete Banach lattice are very important. They lead to the existence of a (positive) spectral measure associated to a SA operator. Consequently, any SA operator has nontrivial invariant subspaces. Obviously, the same property is true for normal operators, in particular for unitary operators. On the other hand, thanks to the fact that constrained extension theorems do work for operators having the bicommutant as a target space, results on the operator-valued Markov moment problem do work too (see O. Olteanu, "Application des théorèmes de prolongement d’opérateurs linéaires au problème des moments et à une généralisation d’un théorème de Mazur-Orlicz." C. R. Acad. Sci. Paris, Série I, 313 (1991), 739-742).

For results related to the "bicommutant" associated to a finite collection of commuting SA operators the interested reader may consult L. Lemnete Ninulescu, "Using the solution of an abstract moment problem to solve some classical complex moment problems", Revue Roumaine de Mathématiques Pures et Appliquées., 51, (2006), 703-711. For the classical definition and related results concerning the bicommutant associated to a single SA operator see R. Cristescu, "Ordered Vector Spaces and Linear Operators", Academiei, Bucharest, and Abacus Press, Tunbridge Wells, Kent (1976).

1. You might also be interested in the Cayley transformation which provides a correspondence between self-adjoint and unitary operators.

2. In many cases the unitary operators form a Lie group (which is, of course, infinite dimensional in infinite dimensional cases).

3. normal operators are the set of operators for which the spectral theorem holds, including unitary operators and self-adjoint operators, as well as positive operators, anti-self-adjoint operators, etc.

The class of normal operators on a Hilbert space have two nice subsets:  the self-adjoint operators and the unitary operators. These two have properties the resembles the real line and the unit circle in the complex plane. [Youngson, Linear Functional Analysis, pg.134]

For a self-adjoint operator, the notion of a positive self-adjoint operator has sense: >=0 for all h in the Hilbert space H. The following result is very useful in applications: for any positive self-adjoint operator A, there exists a unique positive square root of A. Examples of positive self-adjoint operators are: TT* and T*T, where T is an arbitrary linear bounded operator acting on a Hilbert space. In particular, there exist the positive square roots of these two operators.

Unitary operators U (U*=U^(-1)), have important geometric properties. Namely, U and U* preserve the scalar products (hence the norms and the angles), so both of them are isometries.. The spectrum of a unitary operator is contained in the unit circle. Self-adjoint operators have real spectrum. They admit  an integral representation associated to a spectral measure. Normal operators T have the form T = A + i B, with A, B commutting self - adjoint operators. Unitary operators have a similar representation, where A^(2)+B^(2)=I, (AB=BA). Normal and unitary operators admit spectral measures and associated integral representations too.