Setting properties of elements and functions

In[56]:= SetIsometry[s]; 
In[57]:= IsometryQ[s] 
Out[57]= True

In[58]:= SetIsometry[s]; 
In[59]:= aj[s] ** s 
Out[59]= 1

Remember the line In[58] is redundant and unnecessary. Nevertheless, it is restated for the sake of clarity. Normally, In[58] would be left out, because s is still defined as in line In[56].

In[60]:= SetIsometry[s]; 
In[61]:= tp[s] ** s
Out[61]= 1

In[62]:= SetSelfAdjoint[t]; 
In[63]:= tp[t] 
Out[63]= t

In[64]:= SetSelfAdjoint[t]; 
In[65]:= aj[t] 
Out[65]= t

In[66]:= SetProjection[p]; 
In[67]:= p ** p 
Out[67]= p

In[68]:= SetCoIsometry[s]; 
In[69]:= s ** tp[s] 
Out[69]= 1  

In[70]:= SetCoIsometry[w]; 
In[71]:= w** aj[w] 
Out[71]= 1  

In[72]:= SetInv[x]; 
In[73]:= invL[x] 
Out[73]= inv[x]

In[74]:= SetInv[x]; 
In[75]:= invR[x]
Out[75]= inv[x]

In[76]:= SetLinear[f]; 
In[77]:= f[x + y]  
Out[77]= f[x] + f[y]

In[78]:= SetLinear[f]; 
In[79]:= f[(2 + 3 I) x ]  
Out[79]= 2 f[x] + 3 I f[x]

In[80]:= SetConjugateLinear[q]; 
In[81]:= q[x + y]
Out[81]= q[x] + f[y]

In[82]:= SetConjugateLinear[g]; 
In[83]:= g[(2 + 3 I) x ]  
Out[83]= 2 g[x] - 3 I g[x]

In[84]:= SetBilinear[h]; 
In[85]:= h[x + y, w] 
Out[85]= h[x, w] + h[y, w]

In[86]:= SetBilinear[f]; 
In[87]:= f[w, x + y] 
Out[87]= f[w, x] + f[w, y]

In[88]:= SetBilinear[f]; 
In[89]:= f[(2 + 3 I) x, w ] 
Out[89]= 2 f[x, w] + 3 I f[x, w]

In[90]:= SetBilinear[f]; 
In[91]:= f[w,(2 + 3 I) x] 
Out[91]= 2 f[w, x] + 3 I f[w, x]

In[92]:= SetSesquilinear[f]; 
In[93]:= f[x + y, w]  
Out[93]= f[x, w] + f[y, w]

In[94]:= SetSesquilinear[f]; 
In[95]:= f[w, x + y]  
Out[95]= f[w, x] + f[w, y]

In[96]:= SetSesquilinear[f]; 
In[97]:= f[(2 + 3 I) x, w] 
Out[97]= 2 f[x, w] + 3 I f[x, w]

In[98]:= SetSesquilinear[f]; 
In[99]:= f[w,(2 + 3 I) x] 
Out[99]= 2 f[w, x] - 3 I f[w, x]

In[100]:= SetIdempotent[f]; 
In[101]:= f[f[w]]  
Out[101]= w

In[102]:= SetCommutingFunctions[f, g]; LeftQ[f, g] = True; 
In[103]:= g[f[x]] 
Out[103]= f[g[x]]

In[104]:= SetCommutingFunctions[f, g]; LeftQ[f, g] = True; 
In[105]:= f[g[x]] 
Out[105]= f[g[x]]

In[106]:= SetCommutingFunctions[f, g]; LeftQ[f, g] = False; 
In[107]:= g[f[x]] 
Out[107]= g[f[x]]

In[108]:= SetCommutingFunctions[f, g]; LeftQ[f, g] = False; 
In[109]:= f[g[x]] 
Out[109]= g[f[x]]