Regarding to the official docs of angular material the datepicker component provides two output's that get emit if the datepicker is opened (openedStream) or closed (closedStream) (datepicker-api). 
If I open and close the datepicker I get nothing out of those eventsand my <div>{{ message }}</div> remains empty and shows no value.
My actual Setup
export class NoteCreateComponent {
  public message = "";
  
  constructor(){}
  
  toggleDatePicker(picker){
      picker.open();
  }
}
<div>{{ message }}</div>
<button (click)="toggleDatePicker(picker)" mat-icon-button>
<mat-datepicker 
    (openedStream)="message=$event"
    (closedStream)="message=$event" #picker>
</mat-datepicker>
Expected result would either I open or close the datepicker (not only by clicking a button) I get a feedback.
<mat-datepicker 
(opened)="streamOpened()" 
(closed)="streamClosed()" #picker> 
</mat-datepicker>
You should not use openedStream and closedStream event name because they are internal field names of DatePicker component. Use opened and closed. Look at MatDatePicker documetation:
@Output('closed') closedStream: EventEmitter<void>
@Output('opened') openedStream: EventEmitter<void>
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